I want D&D to settle this issue in a way that makes sense, and is good for gameplay. My proposal.
Hit Points
As long as you have at least 1 Hit Point, you can keep fighting. Attacks that hit you cause minor scrapes and bruises, and you fatigue yourself and strain your muscles dodging the worst of the blow. When damage reduces you to 0 HP or below, however, you can no longer resist the strike. Each additional damage causes 1 Wound Point.
When you take any Wound Point damage, make a Constitution save (DC = the amount of WP damage you have). If you fail, you fall unconscious. When your Wound Points equal or exceed your Constitution score, you must make a Constitution save each of your turns. If you fail, you die. Medical care and magical healing can stabilize you so you no longer need to make saves.
Healing
About half of your HP is just stamina. The other half is resilience to pain and minor injuries. Short rests recover stamina, but you can only benefit from two short rests per day. Long rests heal minor injuries, and you can only take one long rest per day.
When you take a short rest of at least five minutes (but no more than twice per day), you can catch your breath and recover one-quarter of your max hit points.
When you take a long rest of at least six hours (but no more than once per day), you can treat your minor injuries and heal one-half your max hit points.
Between these two types of resting, if you get a full day off from adventuring, you can easily heal to full HP. In the course of a typical adventure, though, you might take some damage in a fight, have a short rest, get injured again by a trap, take another short rest, have a major battle, then set up camp and take a long rest. Afterward you might not be at full HP, but if your quest is time-sensitive you won't have time to keep resting.
Wounds
Wound Point damage represents actual broken bones, slashed and punctured flesh, burns, and even brain hemorrhages from psychic attacks. While it is possible to keep fighting after suffering a wound, it's easy for a single additional attack to be fatal, so often you're better-served by heading to safety to treat your wounds, or simply feigning death.
Healing WP damage takes much longer than mere HP damage. You can take an extended rest by spending a whole day in a safe location, during which you cannot exert yourself. Doing so removes 1 Wound Point.
The Heal skill can help recover from WP damage faster, as can magical healing. For instance, the cure light wounds spell removes 1d4 WP. Cure serious wounds lets a character benefit from a short rest and removes 1d6+3 WP. Cure critical wounds lets a character benefit from a long rest and removes 1d8+8 WP.
The rests granted by these spells still count against the normal limit per day. If a character has died in the past five minutes, a cure spell restores them to life and stabilizes them, though they will likely need a very long time to recuperate.