I think there is a more helpful way to look at it, which I will try to explain.The idea of "healing back to full" after a long rest clashes with the idea of hp representing anything more than luck, reserves, bumps and scrapes. This is the crux.
Let's suppose that hit points correspond to some in-game quantity or other - so as hp are marked off on the character sheet, in the fiction the character is being depleted in some fashion; and then as healing occurs, and so hp are added back on to the sheet, the character is restored in respect of that quantity.
What is that quantity?
For obvioius reasons - eg suffering a sword blow can inflict physical injury - whatever the quantity is, it must be the sort of thing that physical injury can deplete.
But for the reason you give, it can't be physical injury - because while physical injuries can be quickly accrued (eg three sword blows in a row), they do not quickly go away. Yet hp can be recovered quickly.
In my view, this is the key: physical injuries trigger events of hp loss, but the hp loss is not itself a measure of physical injury. It's a measure of something that is triggered by physical injury; but which might also be triggered by other things (eg psychic damage); and that can be recovered although the physical injury remains.
What is that "something"? Most simply, although not necessarily that clearly; it's will and capacity to fight. We can break that out a bit: it's stamina, verve, poise, luck, divine favour, etc.
So when your PC suffers a bad fall oir takes a hard blow, and is reduced to zero or near-zero hp, that means that - at that moment - s/hw was nearly all out of luck, and the reserves of stamina and will to go on were depleted. (Think Aragorn at the bottom of the cliff in The Two Towers.)k
When "healing" occurs, that means that your PC's reservs are restored. That healing might be actual physical repair; or it might be a renewal of hope and will. This is why Aragorn, when he remembers Arwen, is able to get back up. He has not healed in any literal sense, but he regains hope and will to struggle on: he gets hp back.
As far as narration is concerned, for the above to work some defntess is needed - it would offfend my own sense of verisimilitude and genre for any amouont of hope to allow someone to get up and carry on despite complete evisceration, for instance. Describing events of hp loss as "blows", "strikes", "cuts", "bruises", etc seems a better fit than pulling out Grey's Anatomy to describe what exactly has been done to which body part.
The main bit of wonkiness the above is likely to lead to in 5e, I think, is that there is an exhaustion track separate from the hit point track. But I think that is going to be wonky whichever narration of hp loss you adopt.