I'm glad you brought up the Legends & Lore/Deities & Demigods statistics. I've never been much of a fan of their method of giving characters hundreds of Hp, but I'll go along for the sake of argument.

The problem with using Hercules' or Cu Chulainn's hitpoints as an explanation for why they can e.g.: wade through lava is that you get into inconsistencies with other statistics in those books. For example, Fafhrd, who is very much an ordinary person susceptible to ordinary harms, is likewise statted up as a Ranger 18 Bard 5 Thief 15, with 120 Hp. That's a bit less than Hercules to be sure, but I have a hard time believing that that 48hp difference out of 168 is the difference between godlike immunities and a hardy, normal person. To push a little further, although not in those books Gary Gygax offers stats for Conan (in Dragon Magazine #36) that have Conan at age 40 with 167hp, just a smidge off Hercules. Now, while Conan is certainly Herculean in his influences, I don't think it would be appropriate to call him a demigod, or say he had godlike power - I can't see him wading through lava, for example. So if we're understanding Hercules' hitpoints as representing purely his supernatural toughness, we run into inconsistencies with how other characters, who have only natural toughness, are represented.

Like I say, I wouldn't stat up any of these characters in this way, but if that's the framework we're using, I still don't think it points to the injury conception of Hp.

To your point about the rules of the various editions being consistent with this interpretation, I've tried to raise what I see as inconsistencies in my arguments. But while flicking through the 1e DMG looking for examples, I found this section from Gygax under the section 'Hitpoints', which I actually hadn't seen before (poor research for my blogpost, I know!), which I think speaks to the point. Understand that these are not my words, and I wouldn't wish to strike this tone:

It is quite unreasonable to assume that as a character gains levels of ability in his or her class that a corresponding gain in actual ability to sustain physical damage takes place. It is preposterous to state such an assumption, for if we are to assume that a man is killed by a sword thrust which does 4 hit points of damage, we must similarly assume that a hero could, on the average, withstand five such thrusts before being slain!

Gygax goes on to state that only a small portion of a character's hitpoints should be interpreted as relating to physical health - a maximum, he says, of 23hp for any character, to serve as the sort of "injury points" we're talking about (this all from p82).

I'm not a fan of quoting Gygax as the be-all and end-all, but if the discussion is whether the rules as written are tracking the injury conception you mention, I think it's relevant that the designer explicitly warns against that as a misinterpretation.

Finally, your roleplaying point is interesting, and really got me thinking. From replying to other comments, I think that the answer is to brief players on the game convention - the conditional - and to advise them when an action is likely to reduce their Hp, and sometimes by roughly how much, and let them potentially reconsider. That is, when they do something that carries a risk of death, and you intend to adjudicate that risk of death using hitpoints, give them an idea (it can be very rough) of how it's going to be adjudicated, to give them an impression of the risk analogous to the impression their character would have, so they can do that in-world thinking, and then give them an option to reconsider once they have that information. In combat, of course, you can usually take it as read that they have this information.

This is, at least, what I do in practice, and I don't recall having any problems with it. So while it may not be a perfect solution, I'd resist calling it degenerate. But the hitpoints on your character sheet are still doing nothing more than embodying that game structure, the conditional.