I see these examples (not just for the bard, but for all classes) as very good ways of showing how those classes exist in the world, what their roles are, and how they fit into the game from a narrative perspective.

That being said, the examples need not represent spellcasting at all. Not all spellcasting requires material components or a focus of some sort. In the examples provided, I can just as easily see those as representations of the bard's class features rather than the bard casting spells. I also don't know that the examples need to exactly match what can be done in game, for the same reason -- they're narrative examples, not gameplay examples. They might also be incomplete examples.

The first example is most likely a demonstration of the bard casting Legend Lore, which has V, S, and M components. The M components for this spell have a specific cost (250gp worth of incense and 4 ivory strips worth at least 50gp each), which means a bardic focus can't be used to cast it, so in this case, if she is casting Legend Lore (and it sounds like it, from the narrative description) there is no bardic focus.

The second and third examples seem to be narrative descriptions of various uses of the bardic class feature Bardic Inspiration which also doesn't require any sort of focus or musical instrument.

To address your specific concerns:

Is it reasonable to assume that RAW or at least RAI that the implication is that in the case of bards, they can use 1) their voice, 2) an improvised musical instrument, or 3) a bought musical instrument?

As far as RAW is concerned, you have one option: a musical instrument. I understand this is vague. The description in Chapter 5, Equipment (5e SRD) says:

Musical Instrument. Several of the most common types of musical instruments are shown on the table as examples. If you have proficiency with a given musical instrument, you can add your proficiency bonus to any ability checks you make to play music with the instrument. A bard can use a musical instrument as a spellcasting focus. Each type of musical instrument requires a separate proficiency.

The table in question is the Tools table, relevant section reproduced below:

\begin{array}{r|ccc} \text{Musical Instrument} & \text{Cost(gp)} & \text{Weight(lbs)}\\ \hline \text{Bagpipes} & 30 & 6 \\ \text{Drum} & 6 & 3 \\ \text{Dulcimer} & 25 & 10 \\ \text{Flute} & 2 & 1 \\ \text{Lute} & 35 & 2 \\ \text{Lyre} & 30 & 2 \\ \text{Horn} & 3 & 2 \\ \text{Panflute} & 12 & 2 \\ \text{Shawm} & 2 & 1 \\ \text{Viol} & 30 & 1 \\ \end{array}

This list is not meant to be exhaustive, of course, and your bard could use any musical instrument that exists in your world. However, the fact that playing a musical instrument has a specific entry in the rules says to me it's more than just banging on something and making noise. The fact the they require proficiency says to me that not just anyone can bang on a drum to make music. I also take it to mean that the bard requires proficiency in the instrument in order to use it as a bardic focus. This is not explicit but it stands to reason, since you need proficiency to 'use' a musical instrument, and a bard must use the instrument as a bardic focus -- this is the RAI part. I have a hard time believing that any designer intended to allow a bard to use a lute as a focus without being able to actually play the lute.

Taking all of this into consideration, I think it would be a stretch to allow a bard to bang on his armor, hum a few bars, or use some other type of improvised musical instrument, and without proficiency, use that as his bardic spellcasting focus.

1. Voice -- No. And even if so, you'd have a hard time saying your verbal components if you're also humming out a few bars as your arcane focus.

2. Improvised instrument -- No. You don't have proficiency in "banging on your armor."

3. A purchased instrument -- Yes. No problem there; it's in the rules.

And if such an assumption isn't RAW/RAI, what are the implications to allowing it as a house rule?

Not too much of a problem here. It's a cool narrative device. Spellcasting foci aren't required ("you can use a [thing] as a spellcasting focus" in every mention of foci in the class rules). It doesn't really affect the bard's ability to cast his spells whether he has a focus or not -- there's always the option of a component pouch -- and not all spells require foci. All in all if you want to allow it there's no real problem; in the end it doesn't make much of a difference.

Ask your DM

Unfortunately, there really isn't any RAW for how to handle damaged tongues or even being mute. For the most part, and except for an optional damage rule from the DMG, there really aren't RAW mechanics for PCs losing body parts.

And all of this is dependent on how a DM defines the disability.

If they rule that the character is mute, then they are effectively under the Silence spell. If it is somewhere in between mute and full speech, then they'll have to make a ruling on how that will affect the Bard.

Workarounds

If the DM does rule that the character is either a mute or their speech is damaged enough to affect Verbal components, then there are some remaining options.

Cast Regenerate

This is the most obvious solution. Either have a party member or a hired caster cast Regenerate to regrow that tongue. Once regrown, all is back to normal.

Dip Sorcerer for Subtle Spell

If regenerating is not an option and they'd like to be able to cast some verbal component spells, then dipping into sorcerer would allow this. However, the downside is needing to forego bard progression in order to do this and this will require a minimum 3 level dip in order to get the Subtle Spell metamagic that would allow bypassing of verbal components.

Simply avoid verbal component spells

This is by far the least attractive option because there are only a few spells under the entire bard list that don't require a verbal component.

This happened at my table, here's what we did and how the players reacted:

In one game I was playing in, this happened to one of the PCs. Our DM ruled that they could not cast a spell with verbal components and they suffered this until we were able to regenerate their tongue.

From a player perspective, the 'maiming' of a PC was awful. It was awful when/why it happened and it was awful for them to have to deal with it for a day until our other cleric could prepare Regenerate. If you are the DM (or anyone reading who is a DM), I would be very wary about mutilating your players and giving them negative mechanical affects. It removes a lot of fun and adds a lot of table issues that can otherwise be avoided.