AngryGM explains this better than I probably could, so instead of going over the "why", I'll just launch into the "what".

First of all, rather than futz about with renaming HP to "fighting spirit", then adding HP back in, then changing hit die to "spirit" die, and all that nonsense, we're going to keep HP as it is. HP is still HP. And specifically, HP is explicitly not meat; it is a combination of things such as luck, stamina, morale, etc. A character who has only lost HP is not actually injured (which is a bit of a problem thematically, since a lot of things that recover HP are fluffed as "healing"). When a character loses all their HP, they don't start dying and bleeding out; instead, they gain the Dispirited Condition:

Dispirited
  • A dispirited creature suffers disadvantage on their attack rolls.
  • Saving throws against the creature's spells and abilities have advantage.
  • The creature gains one level of exhaustion until they are no longer dispirited.
  • A creature that has at least 1 hit point is no longer dispirited.
  • Any damage a dispirited creature takes instead reduces their maximum hit points by that amount.

If this looks familiar, that's because I basically cribbed this from the AngryGM, but added that last bullet point. It is damage to max HP that represents actual injuries. And it does evoke a bit of a death spiral, but not as much in that battle in particular as in future battles before you have a chance to rest. And being injured (i.e. having reduced max HP) affects your stamina and willpower, making you less able to defend yourself (i.e. less HP to tank hits), so it makes sense. Anyway, what this does is it tells the player when it's time to retreat or at least get defensive, and the more damage they take the harder time they'll have in future battles. Dropping to 0 HP isn't "scary", since you're not in danger of dying yet, but it is something that will impair you, and thus players will be incentivized to avoid it as much as they can.

Some other rules for this system:
  • As long as you have at least 1 HP, you won't take any damage to max HP (i.e. no spillover).
  • When max HP drops to 0, then you are bleeding out.
  • If you regain at least 1 max HP, you regain consciousness (just like regaining 1 HP in vanilla).
  • If you stabilize, you regain both 1 HP and 1 max HP after 1d4 hours.

So this leaves the question of how to regain max HP. First, resting. What I'd propose is that you spend hit dice during a long rest to heal max HP the same way you would to heal HP during a short rest. Don't forget that you only regain half your hit dice after a long rest. Spell, especially the ones with words like "heal" or "cure" in their name, might need to be tweaked to restore max HP, and Greater Restoration might also need to be changed so it doesn't just give you back all your max HP (it can remove things like undead curses that reduce max HP, but damage from injuries is different).

This could be a good way to distinguish between Cure Wounds and Healing Word. Perhaps Cure Wounds restores 1 max HP at 1st level, and when upcast it restores an additional 1d4 max HP per spell level. Healing Word, being both a ranged heal and a bonus action to cast, does not restore max HP, at least at 1st level. Perhaps at 3rd, 6th, and 9th level, it restores 1d6, 2d6, and 3d6 max HP, respectively. Heal can either restore 70 HP or split 35/35 between HP and max HP. The paladin's Lay on Hands can use 5 points from the pool to restore 1 max HP. It's kind of messy, so maybe there's some kind of general rule that could be made (e.g. 5 HP of healing can be converted to 1 max HP of healing).

There could also be an added use for the Medicine skill. Perhaps passing a Medicine check on an injured creature that is taking a long rest would give them an extra hit die to roll for healing.

There's also the possibility of taking damage directly to max HP and bypassing your HP. For example, fall damage, or falling into lava. This seems like it would be a tricky thing to manage without it feeling unfair to players, but it makes logical sense that such things would injure you directly (you can't dodge lava).

Anyway, what are your thoughts on this? Don't forget to read the AngryGM article, as he goes into much greater depth as to why he wanted to do something like this. Do you prefer his version? What do you see as improvements (or not) of my version over his?