Are the Book of Ancient Secrets level One ritual spells automatically capped at Two?

Question

At level 3, Warlocks with the Book of Ancient Secrets eldritch invocation would be able to start collecting ritual spells from all the other magic classes:

Choose two 1st-level spells that have the ritual tag from any class’s spell list (the two needn’t be from the same list). The spells appear in the book and don’t count against the number of spells you know. With your Book of Shadows in hand, you can cast the chosen spells as rituals. You can’t cast the spells except as rituals, unless you’ve learned them by some other means. You can also cast a warlock spell you know as a ritual if it has the ritual tag.

On your adventures, you can add other ritual spells to your Book of Shadows. When you find such a spell, you can add it to the book if the spell’s level is equal to or less than half your warlock level (rounded up) and if you can spare the time to transcribe the spell. For each level of the spell, the transcription process takes 2 hours and costs 50 gp for the rare inks needed to inscribe it.

Since a Warlock has to be level 3 and have the Pact of the Tome, am I correct in my understanding that they are unable to learn any further level 1 rituals, other than the original two spells picked when first acquiring the invocation?

To be clear, my confusion is regarding the bold text in the invocation above. If you can only learn new rituals of your warlock level or less than half rounded up, then at level 3, I would only be able to learn level 2 and level 3 rituals spells. As half of three is 1.5, rounded up = 2.

So, if a warlock can only access the Book of Ancient Secrets at level 3, and you can only learn rituals at your level or less than half rounded up, are you limited to just the two first level ritual spells upon taking the invocation of Book of Ancient Secrets.

Answer 1

No. He starts with those two when he takes the invocation at Level 3. But he can copy other spells of the Ritual Casting type if he finds them.

On your Adventures, you can add other ritual Spells to your Book⁠ of Shadows. When you find such a spell, you can add it to the book if the spell’s level is equal to or less than half your Warlock level (rounded up) and if you can spare the time to transcribe the spell.

He needs only pay the gold and spend the time. Equal to or less than means spells of level 2 or 1 can be added.

Answer 2

Your confusion mostly seems to stem from inaccurately parsing part of the English language. I don't necessarily blame you for that since English is often significantly ambiguous, and this is one of those circumstances.

The way you seem to have taken the sentence is "spells of level equal to your Warlock level, and spells of level less than half your Warlock level". Mathematically, this would actually mean only Level 1 and Level 3 spells, since 2 is not less than 2.

However, that isn't the most natural interpretation. The "equals" half ends up referring to the entire phrase just like the "less than" half, and instead you get "spells of level equal to half your Warlock level, and spells of level less than half your Warlock level." So for a Level 3 Warlock, you can copy spells equal to level 2 or less than level 2 (a.k.a. level 1), but not level 3. This never leaves "gaps" where a spell could have been learned if its level were higher, and level 1 spells are always valid.

Answer 3

Your edit heavily changes the question.

if the spell’s level is equal to or less than half your warlock level (rounded up)

Spell's Level <= (warlock level / 2) rounded up.
So if your warlock level is 3, half of that is 1.5.
Rounded up that is 2.
You can learn spells that are equal to (2) or less than (1).
You cannot learn a level 3 spell at that stage, as it is GREATER than half your warlock level rounded up.

Answer 4

There are multiple ways a warlock can learn new spells and Book of Ancient Secrets rituals.

First, the Book of Ancient Secrets provides a way for you to learn ritual spells NOT as part of your class feature, but as part of the invocation (this is NOT part of the Pact Magic feature and can spells from any written source)

On your adventures, you can add other ritual spells to your Book of Shadows. When you find such a spell, you can add it to the book if the spell's level is equal to or less than half your warlock level (rounded up) and if you can spare the time to transcribe the spell. For each level of the spell, the transcription process takes 2 hours and costs 50 gp for the rare inks needed to inscribe it. Book of Ancient Secrets PHB p. 110

Also, there's learning Warlock (and subclass) spells as part of your Pact Magic (Since the Book of Ancient secrets allows you to cast any spell you know that has the ritual tag as a ritual). A warlock can learn new spells each time they level up. There is no requirement that they are of the current spell slot level.

The Spells Known column of the Warlock table shows when you learn more warlock spells of your choice of 1st level and higher. A spell you choose must be of a level no higher than what's shown in the table's Slot Level column for your level. When you reach 6th level, for example, you learn a new warlock spell, which can be 1st, 2nd, or 3rd level. Pact Magic PHB p.107

You can also switch out a currently known spell for another spell

Additionally, when you gain a level in this class, you can choose one of the warlock spells you know and replace it with another spell from the warlock spell list, which also must be of a level for which you have spell slots. Pact Magic PHB p.107