No, it doesn't "shift" the problem. It still requires some adjudication in terms of the specifics of what's being represented, but it still establishes what's being represented. That's a far smaller cognitive burden than having to determine if it's an injury or just some sort of depletion of personal stamina, luck, divine protection, etc.
You say it's different. I say that it's the same. The loss of HP establishes what's being represented per the game's understanding of HP, but it still requires adjudication in terms of the specifics of what's being represented.
Which is why I've noted previously that the game is more about heroic fantasy than realism, with characters taking numerous injuries (oftentimes serious) with no corresponding loss of prowess. In that regard, you're free to narrate any particular set of wounds that you want (within reasonable limits, obviously; no one is going to seriously countenance you saying someone's head has been chopped off or their eyes gouged out) without creating any sort of cognitive gap in doing so, unlike the one that you'd get when you suffer burn wounds and then have someone else at the party make them go away (according to the game's operations) by shouting at you to buck you up.
Again, the game is silent about whether your burns go away. You choose to read "heal" to mean that "your burns go away," but in the context of 4e rules, we are told that healing only represents the regaining of HP, which are a variety of aspects of a character's combat vigor. Nothing more.
More convenient for you, certainly, but not better.
And it "helps them heal."
You are again losing sight of the forest for the trees. I have already referred you to the rules of healing and what they and HP represent in the context of 4e, so "helps them heal" is not the gotcha that you think that it is. I think that this is a consistent problem with your reading of 4e. You read it in isolated pieces. You don't try to understand the rules in the context of the big picture. You have little to no actual game experience with 4e.
No, it's the game's hang-up, too. That's fairly self-evident from the fact that the mechanical operation is exactly the same as the cleric's healing word (4E PHB p. 62), in that they use positive energy to let the target spend one of their own healing surges, while the text describes it as "You whisper a brief prayer as divine light washes over your target, helping to mend its wounds." Given that it wants to have the same thing be two different things (i.e. "recover resiliency" and "mend wounds"), that creates an issue with what the keyword is denoting, since it can vary by context, and so that context has to be parsed by the players (including the DM) more than they would if it was only connoting one thing.
@pemerton addressed this already.
More like a self-evident truth. Having one mechanic present potentially two different things depending on the circumstance is more complex than if one mechanic presented itself as only one thing, period. That's not really something that can be argued. You might say that's not a big deal, and it might not be for you, but it's undeniably more complicated (and doesn't need to be).
One mechanic doesn't represent just two different things. It's not just wounds and everything else. In D&D HP represents a variety of elements that factor into a character's survival. Wounds are one among several others. Is that "more complex than if one mechanic presented itself as only one thing"? If so, it appears to be negligibly so for most people who are playing and enjoying the game. So despite what you would assert here, I would say that it is deniably more complicated, even if we were to agree that it was "more complicated."
That's a semantic distinction, and not really helpful in what it offers. If something is vague, then you have to expend mental energy figuring it out. It's a gap that your cognition bridges. Changing the shorthand doesn't change that.
You may call it a semantic distinction, but I do think that it is helpful in that vagueness offers more explanatory power and is a more readily apparent problem that we regularly deal with in communication with little fuss in our daily lives. I think that "cognitive gap" tries to present a simple case of vagueness as a much more insurmountable technical problem of cognitive incomprehensibility than it actually is.
For comparison, vagueness of rules is lauded by some as a "feature" in 5e that empowers GMs to make rulings rather than be beholden to rules.
Except the game used to get alone just fine doing exactly what I've been talking about.
It sounds like you are in danger of conflating yourself with the game here.
Remember, 4E changed things up by actually having that idea of "Hit point loss/restoration isn't just injury" actually be present in the game's mechanical operations. Prior to that, the idea received lip service in essays, but wasn't ever actually suggested in how the mechanics functioned. Then a change was made, and while that might have addressed certain issues, it also presented new ones, which to my mind were much greater than anything it fixed.
What this tells me is that 4e is internally consistent in word and deed with how the mechanics and operations of HP are reflected in the fiction. Way to go, 4e!
No, I disagree strongly, in that 4E took its gamist aspects much further than any edition before or since, and so abandoned much of the heavy lifting with regard to conveying what the various operations were actually representing from an in-game standpoint.
As you yourself noted, it's vague.
I said that your complaint involves an issue of vagueness about HP rather than an inherently problematic cognitive gap. I was not, however, saying that 4e was vague. I think that 4e is quite the opposite. It's quite explicit. 4e powers tell us a LOT about the fiction. The designers were wholly transparent and not one bit shy about what the mechanical processes, terms and keywords, and various operations of the game represent from an in-game standpoint, and it was fairly consistent in that regard. Things may not map perfectly (e.g., square fireballs) but we are told what the mechanics are meant to represent in the fiction.
From a gamist standpoint; not in terms of actually connecting meta-game operations to in-character happenings.
Please stop using the term "gamist" for things that you dislike or go against your own sense of self-proclaimed simulationism, because powers are very forthright in telling you what the meta-game operations represent for in-character happenings. You may not like what they are telling you they represent or are simulating, but they are telling you. There is a difference between those two positions, and it's not even a subtle one.
You don't "have" to deal with the problem, in the sense that you can just ignore the issue. Or simply bridge the cognitive gap on your own. In that regard, it's much like any other issue, in that you don't have to fix a broken stair when you can just step over it, or you might think that fixing the stair isn't a very hard job to do, and can be accomplished with ease. But I'd say it's still better for the stair not to be broken in the first place.
We can't even agree that the stair is broken or not! That's the problem. You asserting the existence of a problem and a cognitive gap that others either don't see as being problem or believe exists at all. You are saying "this is broken and needs fixing!" Others are telling you, "this isn't broken and it's working as intended." If a cognitive gap exists, it's between these two positions more so than anything.
I'd like to encourage you to read the rules as well, so that you can have a better appreciation for things like how the framing fiction for inspiring word says it restores the same hit points that a fireball causes, despite the fact that one is causing fire damage and the other is reinvigorating you, making the players (including the DM) be the ones who have to track which thing the hit point changes are representing (i.e. injury or stamina) when mapping the game-play to what's happening in the setting.
I know that you like to turn things back on people in "no you!" games, but I've read and played the rules of 4e. In contrast, I remember that you were surprised in one thread (or maybe here) about the existence of 4e healing surges and their associated rules when people told you about them.
I don't think that healing spells or powers restore "the same" HP that was lost by a fireball or a sword or by a monster bite. I think that HP can be restored by different sources and through different means. I think that it has never been as clear cut as losing 20 HP to a fireball and having a healing spell erasing 10 damage from those burn wounds. Do my burn wounds disappear within 24 hours just because I sleep in 5e? Do my burn wounds disappear when I take a short rest and spend HD to heal myself in 5e? I don't think that this is what these operations are meant to represent in the fiction. The same is true in regards to a warlord's abilities.
But to reiterate, the problem is that the game indicates that the same operation can represent multiple different things, creating a cognitive gap where the players (including the DM) have to then figure out how to connect the two.
You keep asserting and taking for granted in your argumentation the creation, existence, and problematic nature of this "cognitive gap" regarding HP, but I don't think that you are doing a good job of demonstrating these things in your argument, which comes across as a series of compounded assumptions. Does this cognitive gap exist? Is this phenomenon what creates it or is it something else? How is this even a problem?
If your character is badly burned, and had a warlord yell at them, then there's an issue with figuring out how they keep taking wounds and not being healed, but rather motivated to stay active even when the "wound hp" damage they've taken exceeds their total hp, but the numbers are still in positive amounts thanks to "resilience hp"-type healing.
Let's be clear here. You have an issue with it. You don't know how to reconcile these two things in accordance with your own idiomatic preferences. I and others don't necessarily share agreement that this "issue" exists. There is nothing for us to reconcile as it's perfectly consistent with our understanding of what the mechanics represent in the fiction.
@pemerton and I have already shared our readings of this power and what this mechanic represents in the fiction, using the rules and text of 4e to do so even. So there is little point repeating myself here just because you choose to do so.
It's actually less clear than in other editions; all of them, in fact. That's because the gamist applications that 4E champions simply don't prioritize the connecting of the mechanics with in-game representation. If they did, we wouldn't have Schrodinger's hit points, where they can be injury one round and then personal stamina the next.
Except they aren't Schrödinger's HP - which is hardly an apt use of Schrödinger's Cat - because the understanding of HP is consistently applied in 4e as being a variety of factors that contribute to the character's survival. 4e is loud and clear about its HP and what they represent.
In fact, I think that it's the vagueness of HP in other editions that you liked, because it was that vagueness regarding HP and associated operations that empowered you to read HP in terms of your stated preferences.