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.
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.
Improvised instrument -- No. You don't have
proficiency in "banging on your armor."
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.
Bards don't prepare spells in D&D 5th edition.
The Players Handbook, chapter 10: Spellcasting, mentions that Bards learn spells and can cast them spontaneously. The section "Known and Prepared Spells" says the following:
Before a spellcaster can use a spell, he or she must have the spell firmly fixed in mind, or must have access to the spell in a magic item. Members of a few classes, including bards and sorcerers, have a limited list of spells they know that are always fixed in mind.
Instead of preparing spells, Bards simply have spells known, and they can expend spell slots to cast one of their known spells. When creating a 1st level Bard, check the PHB for the Bard's spellcasting feature, under the subsection "Spells Known of 1st Level and Higher":
You know four 1st-level spells of your choice from the bard spell list.
The Spells Known column of the Bard table shows when you learn more bard spells of your choice. Each of these spells must be of a level for which you have spell slots, as shown on the table. For instance, when you reach 3rd level in this class, you can learn one new spell of 1st or 2nd level.
Additionally, Bards learn a new spell (and can swap out an old spell) whenever they gain another level in the Bard class. You can read the full text of their Spellcasting feature on D&DBeyond.
Best Answer
“D&D 5e does not differentiate between fluff and crunch,” they always say. “You need a finger to cast fireball,” they insist. Except when it’s a flameskull.
The answers that are going to be the most upvoted are the ones that state it only matters whether the spell has a verbal component, because that’s how most people play; because nobody (except you) thought of that when reading the bard’s class description – because that’s what most people feel is right. Even if it isn’t – that has no bearing on votes.
So in spite of the votes, you might actually be right that that is what the text implies (RAW). However, I am under the impression that many people on the Internet care a lot (too much, see below) about the text (the RAW), so you might find them arguing that that is not what the text is stating, because otherwise the way they’ve been thinking about and playing the game all this time was “wrong,” which they couldn’t accept.
With all the arguing about the RAW, the following seems overdue to me:
Frame challenge: Does it matter?
Suppose the interpretation you suggested is correct, i.e., the words in the book mean what you think they mean. Does it matter that we made a “mistake” when we allowed a bard to cast counterspell in a field of silence? Does it matter that we are playing the game “wrong” if we keep allowing it? No. What matters is whether we’re having fun.
I remember Matt Colville talking about how a non-D&D-player watched a session of 3.5e and described it as something like “20 minutes of fun packed into four hours.”¹ Because the rules of 3.5e were so comprehensive, because there was a rule for virtually everything, there was a lot of time spent on looking them up. But that’s not fun. That’s why D&D 5e leaves so many things open that 3.5e had rules for: So that the DM can make a decision on the spot and people can go back to playing – to having fun.
You know what’s also not fun? Rules that read as though they were written by a lawyer. Who would wanna read those? That’s why 5e’s rules are written in simple terms that you can immediately understand. What if those wordings don’t cover all cases or have nonsensical consequences upon closer inspection? Who cares?! The DM can always just say: “That’s stupid, we’re doing it this way.”
For example, imagine you are playing a variant human wizard and you chose Lightly Armored as your starting feat. As you leveled up, you picked up Moderately Armored and Heavily Armored. Now your character dies and is resurrected as a non-human (non-dwarf) by your party’s druid with a reincarnate spell. That means you lose your human traits, including your Lightly Armored feat. Because you now no longer meet the prerequisite of the Moderately Armored feat (proficiency with light armor), you lose the benefits of that feat (PHB, p. 165), and with that, Heavily Armored is rendered useless as well. – That is what the rules require. But is any (good) DM going to insist on that? It is more likely that they will simply have you lose Heavily Armored. Or they might allow you to keep the variant human feat in exchange for some of the traits of the race you assumed. Or maybe they’d just allow you to keep the feat, at no cost. Why ruin your fun by taking two of your feats away? “Because that’s what the rules say”? – “Because that’s what my character would do”?
People on the Internet seem to care a lot about the RAW; about whether you need a finger to cast fireball, and whether flameskulls are exempt from that. The truth is (so I assert), if you believe that the rules require you to have a finger to cast fireball, then you must also accept that they consequently prohibit flameskulls from doing so, despite it being listed as one of their spells. But that does not matter. Whether or not you can cast fireball without fingers depends on whether your DM will allow it (should that ever come up), and they will probably let a flameskull cast it either way. You should think less about the letter of the rules and more about how you could have more fun at the table.
Q: But what about the RAI?
A: It is not intended, I am certain. If it were, it would be stated much more explicitly in the rules. You are also quoting a section that is clearly intended as flavor text. Even if people insist that’s not a thing, the authors wouldn’t sneak mechanically relevant information into such passages. Compare the fact that ki is described as “magical energy” in such a section (PHB, p. 76) and yet has been clarified by Jeremy Crawford to not be “defined as magical for game purposes.”
Q: Are you really saying we don’t need rules?!
A: Not in the rigorous manner other games need them, such as Magic: The Gathering. That’s a competitive game, so it needs clear rules – getting into an unending argument because the rules are ambiguous or having a game decided by an arbitrary-seeming judge ruling is not fun. D&D, however, is not a competition. We need rules to the extent that they facilitate play – that they facilitate fun – and that stops somewhere. It stops way before Fingers & Flameskulls. Don’t obsess over them.
¹ [If somebody finds it, please edit this answer and put the link here. Maybe also change the wording to “Matt Colville said in one of his videos…”]