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Now that we have settled on this, let's move to the next question: what is a draconequus?

Later, this starts to fade as Discord's second nature become more evident: Chaos. Once again Equestria seems to parallel Origin's Ultima. Chaos doesn't seems to be inherently evil but just a force of nature closely related to entropy. "Chaotic" Discord only want to have fun and subvert things, but once reformed and becoming friend with Fluttershy he is perfectly able of being chaotic while also being a friend that likes to laugh with others, not at others. In this regard it should also be noticed that Accord, in his misguided attempts, seems to instead see (force) Harmony as a way to reach pure Order. This should not surprise you, since Accord in that story acts as a Bane of Order, and thus is taking the nature of Order to its extreme (this is similar to Ei in Genshin Impact trying to create immutable Eternity by removing people's ambitions).

At this point we can conclude one thing: Discord actual nature is that of a creature of Chaos and Chaos alone, and while he used to think that Chaos should equal to Disharmony the story shows that is not necessary the case.

We can see this clearly in "Discordant Harmony, S7E12": in the episode Discord tries to act normally and suppress his Chaotic nature for Fluttershy sake thinking his nonsensical ways would annoy her (while actually she didn't care at all in the first place). But in doing so he almost destroys himself.

Notice how "acting normal" can harm Discord, yet being friendly with Fluttershy causes him no harm. It is unclear how things changed in development (Discord's name seems to indicate the staff initially intended chaos and disharmony to be the same thing) but as the show progressed Discord is presented simply as a creature of Chaos that choose Friendship over Disharmony.

Discord is a creature of Chaos and Entropy that takes a form that represent that Chaos. Before being reformed he had almost no friends (well, apparently there was a big green blob) and he choose to be a pawn of Disharmony, thinking that Chaos should subvert friendship (like his cousin thinks that Chaos should bring fear). Later, a yellow pony showed him that was not the case and that he could just be a (sometime annoying) prankster that loves been nonsensical and chaotic.

Now that we have settled on this, let's move to the next question:

What is a draconequus?

Later, this starts to fade as Discord's second nature become more evident: Chaos. Once again Equestria seems to parallel Origin's Ultima. Chaos doesn't seem to be inherently evil but just a force of nature closely related to entropy. "Chaotic" Discord only want to have fun and subvert things, but once reformed and becoming friend with Fluttershy he is perfectly able of being chaotic while also being a friend that likes to laugh with others, not at others. In this regard it should also be noticed that Accord, in his misguided attempts, seems to instead see (force) Harmony as a way to reach pure Order. This should not surprise you, since Accord in that story acts as a Bane of Order, and thus is taking the nature of Order to its extreme (this is similar to Ei in Genshin Impact trying to create immutable Eternity by removing people's ambitions).

At this point we can conclude one thing: Discord's actual nature is that of a creature of Chaos and Chaos alone, and while he used to think that Chaos should be equal to Disharmony the story shows that is not necessary the case.

We can see this clearly in "Discordant Harmony, S7E12": in this episode Discord tries to act normally and suppress his Chaotic nature for Fluttershy's sake, thinking his nonsensical ways would annoy her (while actually she didn't care at all in the first place). But in doing so he almost destroys himself.

Notice how "acting normal" can harm Discord, yet being friendly with Fluttershy causes him no harm. It is unclear how things changed in development (Discord's name seems to indicate the staff initially intended chaos and disharmony to be the same thing) but as the show progressed Discord is presented simply as a creature of Chaos that chooses Friendship over Disharmony.

Discord is a creature of Chaos and Entropy that takes a form that represent that Chaos. Before being reformed he had almost no friends (well, apparently there was a big green blob) and he chose to be a pawn of Disharmony, thinking that Chaos should subvert friendship (like his cousin thinks that Chaos should bring fear). Later, a yellow pony showed him that was not the case and that he could just be a (sometime annoying) prankster that loves being nonsensical and chaotic.

At this point we can conclude one thing: Discord is a creature of Chaos that was misguided in thinking he needed Disharmony but does need to.

In "Discordant Harmony, S7E12" Discord tries to act normally and suppress his Chaotic nature for Fluttershy. But in doing so he almost destroys himself.

Notice how "acting normal" can harm Discord, yet being friendly with Fluttershy causes him no harm. It is unclear how things changed in development (Discord's name seems to indicate the staff initially intended chaos and disharmony to be the same thing) but at the end Discord is just a creature of Chaos.

Discord is a creature of Chaos and Entropy that takes a form that represent that Chaos. During his life he had a few friends and become a pawn of Disharmony, thinking that Chaos should subvert friendship. Later, a yellow pony showed him that was not the case and that he could just be a (sometime annoying) prankster that loves been nonsensical and chaotic.

At this point we can conclude one thing: Discord actual nature is that of a creature of Chaos and Chaos alone, and while he used to think that Chaos should equal to Disharmony the story shows that is not necessary the case.

Discord can choose Harmony and Friendship over Disharmony and strife, but he is unable to refuse his own Chaotic nature.

We can see this clearly in "Discordant Harmony, S7E12": in the episode Discord tries to act normally and suppress his Chaotic nature for Fluttershy sake thinking his nonsensical ways would annoy her (while actually she didn't care at all in the first place). But in doing so he almost destroys himself.

Notice how "acting normal" can harm Discord, yet being friendly with Fluttershy causes him no harm. It is unclear how things changed in development (Discord's name seems to indicate the staff initially intended chaos and disharmony to be the same thing) but as the show progressed Discord is presented simply as a creature of Chaos that choose Friendship over Disharmony.

Discord is a creature of Chaos and Entropy that takes a form that represent that Chaos. Before being reformed he had almost no friends (well, apparently there was a big green blob) and he choose to be a pawn of Disharmony, thinking that Chaos should subvert friendship (like his cousin thinks that Chaos should bring fear). Later, a yellow pony showed him that was not the case and that he could just be a (sometime annoying) prankster that loves been nonsensical and chaotic.

As a character, Discord original design came straight from Lauren Faust herself, which in turn took inspiration from the character Q in Star Trek (and this should explain why they got De Lancie himself to voice him, see here for further details).

Jokes aside, it is pretty clear that a simple "dragon horse" wasn't what she had in mind. As you can see in this concept art kindly taken from her DeviantArt account, Discord was planned having multiple animals traits from the start.

We first hear the term from Cheerilee in "The Return of Harmony Part 1 - S2E1"

Cheerilee: This creature is called a draconequus. He has the head of a pony and a body made up of all sorts of things. What do you suppose that represents? Apple Bloom: Confusion!
Sweetie Belle: Evil!
Scootaloo: Chaos!
[...cut...]
Cheerilee: Actually, in a way, you're all right. This statue represents discord, which means a lack of harmony between ponies.

• first she describes the feature of a draconequus as "made up of all sorts of things" - not just Discord
• yet later she also describes a draconequus as a representation of "disharmony" It is also worth noticing that Scootaloo answer was "Chaos", yet based on Cheerilee answer disharmony and chaos are two separate things.

As you can see when Discord was first introduced, he was described as the draconequus embodiment of "disharmony". This is reinforced by the fact that the scene shows the seal on Discord breaking as ponies gets in an argument near the statue. This scene is telling us that Discord is empowered by the three Cutie Mark Crusaders fight.

Later, in the same episode, Celestia again calls Discord a creature of disharmony

It is only in the next episode than Discord defines himself a creature of Chaos.

Twilight Sparkle: Stop it, Discord. You're not playing fair. Discord: I'm not playing fair? Perhaps we haven't met. I'm Discord, spirit of chaos and disharmony.

At this point, some small incongruousness start to come up.

Are all draconequus really "embodiment of disharmony?"

We never see another draconequus in the series - the only one is Discord. Therefore, we can't really argue if "other" draconequus are like him simple because we don't know any.
Yet it is worth noticing that Accord - Discord alter-ego in "My Little Pony: Friendship is Magic Issue #48-50 story arc Chaos Theory" - is both a creature of pure Order and still a draconequus. Therefore, it doesn't seem likely the two things are directly tied.

Are all "embodiment of disharmony" draconequus?"

Princess Eris, first introduced in "My Little Pony: Nightmare Knights" as Discord's cousin in the goddess of Chaos of an alternate universe (yes, don't overthink how this seems oddly similar to a certain "Cat of Destruction" and his brother...) is... not a draconequus as we know and never uses the word. If anything she is more close to a red parrot, or to be more precise a Sarimanok (which does indeed mean "assorted bird" so there is a strong possibility someone took that as "bird made up of different parts" - a parallel to the draconequus).
This proves that not every "creature of Chaos and Disharmony" has to take the shape of a draconequus. Some take a step further and claims that this is evidence of Discord current appearance being something he choose.

At first, Discord main nature is that of "Disharmony" - lack of friendship, the main show theme. We can se it in his name - Discord comes from "Discordia", the Latin name of Eris, the goddess of discord. And as you just saw, Discord's cousin is called Eris.
In is first appearance we see Discord's Disharmony side. He acts as a malevolent creature that takes power from discord and strife. He enjoys subverting the mane six nature and turn them into their opposite - what some fans call the "Element of Disharmony". He is actively evil and wants to do harm.

Later, this start to fade as Discord's second nature become more evident: Chaos. Once again Equestria seems to parallel Origin's Ultima. Chaos doesn't seems to be inherently evil but just a force of nature closely related to entropy. "Chaotic" Discord only want to have fun and subvert things, but once reformed and becoming friend with Fluttershy he is perfectly able of being chaotic while also being a friend that likes to laugh with others, not at others. In this regard it should also be noticed that Accord in his misguided attempts seems to instead see (forced) Harmony as a way to reach pure Order. This should not surprise you, since Accord in that story acts as a Bane of Order, and thus is taking the nature of Order to its extreme (this is similar to Ei in Genshin Impact trying to create immutable Eternity by removing people ambitions).

At this point can conclude one thing: Discord is a creature of Chaos that was misguided in thinking he needed Disharmony but does need to.

Notice how "acting normal" can harm Discord, yet being friendly with Fluttershy cause him no harm. It is unclear how things changed in development (Discord name seems to indicate the staff initially intended chaos and disharmony to be the same thing) but at the end Discord is just a creature of Chaos.

So, to answer your question..

Discord is a creature of Chaos and Entropy that take a form that represent that Chaos. During his life he had few friends and become a pawn of Disharmony, thinking that Chaos should subvert friendship. Later, a yellow pony showed him that was not the case and that he could just be a (sometime annoying) prankster that loves been nonsensical and chaotic.

As a character, Discord's original design came straight from Lauren Faust herself, which in turn took inspiration from the character Q in Star Trek (and this should explain why they got De Lancie himself to voice him, see here for further details).

Jokes aside, it is pretty clear that a simple "dragon horse" wasn't what she had in mind. As you can see in this concept art kindly taken from her DeviantArt account, Discord was planned as having multiple animals traits from the start.

We first hear the term from Cheerilee in "The Return of Harmony Part 1 - S2E1":

Cheerilee: This creature is called a draconequus. He has the head of a pony and a body made up of all sorts of things. What do you suppose that represents? 
Apple Bloom: Confusion!
Sweetie Belle: Evil!
Scootaloo: Chaos!
[...cut...]
Cheerilee: Actually, in a way, you're all right. This statue represents discord, which means a lack of harmony between ponies.

• first she describes the feature of a draconequus as "made up of all sorts of things" - not just Discord
• yet later she also describes a draconequus as a representation of "disharmony"

It is also worth noting that Scootaloo's answer was "Chaos", yet based on Cheerilee's answer disharmony and chaos are two separate things.

As you can see when Discord was first introduced, he was described as the draconequus embodiment of "disharmony". This is reinforced by the fact that the scene shows the seal on Discord breaking as ponies get in an argument near the statue. This scene is telling us that Discord is empowered by the three Cutie Mark Crusaders' fight.

Later, in the same episode, Celestia again calls Discord a creature of disharmony:

It is only in the next episode than Discord defines himself as a creature of Chaos.

Twilight Sparkle: Stop it, Discord. You're not playing fair. 
Discord: I'm not playing fair? Perhaps we haven't met. I'm Discord, spirit of chaos and disharmony.

At this point, some small incongruities start to come up.

Are all draconequus really an "embodiment of disharmony"?

We never see another draconequus in the series - the only one is Discord. Therefore, we can't really argue if "other" draconequus are like him, simply because we don't know any.
Yet it is worth noticing that Accord - Discord's alter-ego in "My Little Pony: Friendship is Magic Issue #48-50 story arc Chaos Theory" - is both a creature of pure Order and still a draconequus. Therefore, it doesn't seem likely the two things are directly tied.

Are all "embodiments of disharmony" draconequus?

Princess Eris, first introduced in "My Little Pony: Nightmare Knights" as Discord's cousin in the goddess of Chaos of an alternate universe (yes, don't overthink how this seems oddly similar to a certain "Cat of Destruction" and his brother...) is... not a draconequus as we know and never uses the word. If anything she is more close to a red parrot, or to be more precise a Sarimanok (which does indeed mean "assorted bird" so there is a strong possibility someone took that as "bird made up of different parts" - a parallel to the draconequus).
This proves that not every "creature of Chaos and Disharmony" has to take the shape of a draconequus. Some take a step further and claims that this is evidence of Discord's current appearance being something he chose.

At first, Discord's main nature is that of "Disharmony" - lack of friendship, the main show theme. We can se it in his name - Discord comes from "Discordia", the Latin name of Eris, the goddess of discord. And as you just saw, Discord's cousin is called Eris.
In his first appearance we see Discord's Disharmony side. He acts as a malevolent creature that takes power from discord and strife. He enjoys subverting the mane six's nature and turns them into their opposites - what some fans call the "Element of Disharmony". He is actively evil and wants to do harm.

Later, this starts to fade as Discord's second nature become more evident: Chaos. Once again Equestria seems to parallel Origin's Ultima. Chaos doesn't seems to be inherently evil but just a force of nature closely related to entropy. "Chaotic" Discord only want to have fun and subvert things, but once reformed and becoming friend with Fluttershy he is perfectly able of being chaotic while also being a friend that likes to laugh with others, not at others. In this regard it should also be noticed that Accord, in his misguided attempts, seems to instead see (force) Harmony as a way to reach pure Order. This should not surprise you, since Accord in that story acts as a Bane of Order, and thus is taking the nature of Order to its extreme (this is similar to Ei in Genshin Impact trying to create immutable Eternity by removing people's ambitions).

At this point we can conclude one thing: Discord is a creature of Chaos that was misguided in thinking he needed Disharmony but does need to.

Notice how "acting normal" can harm Discord, yet being friendly with Fluttershy causes him no harm. It is unclear how things changed in development (Discord's name seems to indicate the staff initially intended chaos and disharmony to be the same thing) but at the end Discord is just a creature of Chaos.

So, to answer your question...

Discord is a creature of Chaos and Entropy that takes a form that represent that Chaos. During his life he had a few friends and become a pawn of Disharmony, thinking that Chaos should subvert friendship. Later, a yellow pony showed him that was not the case and that he could just be a (sometime annoying) prankster that loves been nonsensical and chaotic.