Does a Swords Bard's extra damage count as damage that the original attack inflicts for Vampiric Bite's HP gain and roll bonus?

Question

I know that Slashing Flourish, and the rest of the options, deal extra damage to the original target, but I'm asking if the damage it deals to creatures of your choice within 5 feet of you counts as damage from the attack.

Slashing Flourish. You can expend one use of your Bardic Inspiration to cause the weapon to deal extra damage to the target you hit and to any other creature of your choice that you can see within 5 feet of you. The damage equals the number you roll on the Bardic Inspiration die.

I am mainly asking this to see if it qualifies for the Dhampir's vampiric bite feature

When you attack with this bite and hit a creature that isn’t a Construct or an Undead, you can empower yourself in one of the following ways of your choice:

• You regain hit points equal to the piercing damage dealt by the bite.
• You gain a bonus to the next ability check or attack roll you make; the bonus equals the piercing damage dealt by the bite.

Answer 1

Yes, the vampiric bite fulfills the requirement for Slashing Flourish and viceversa.

The bard's feature requires that you take the Attack action and that you make a weapon attack.

The description of vampiric bite says:

Your fanged bite is a natural weapon, which counts as a simple melee weapon with which you are proficient.

The Vampiric Bite and the Slashing Flourish can interact.

In particular, Slashing Flourish says (emphasis mine):

You can expend one use of your Bardic Inspiration to cause the weapon to deal extra damage [...]

hence the extra damage is caused by the weapon you are using for the attack. When you activate the bite's feature that allows to regain hit points, you regain a number of HPs equal to the total damage dealt, normal plus the bonus one given by the Slashing Flourish.

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From a thematic point of view, the wording and the description of College of Swords (not to mention the scimitar proficiency) suggests that the flavour of this subclass's features is about swords. Anyway, there is nothing in the rules that denies to use these features with other suitable weapons: it is up to the player and to the DM to describe how they can be adapted.

For example, since another feature of Slashing Flourish is to damage other creatures beside the target of the attack, you can describe your bite Attack Action using Slashing Flourish as a ravaging attack, furiously biting and hitting nearby creatures.

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From the Sage Advice Compendium, natural weapons are weapons.