I'm most familiar with D&D 3.5e and 5e, which both have pretty similar ways of describing hit points. The 3.5e SRD says
Hit points mean two things in the game world: the ability to take physical punishment and keep going, and the ability to turn a serious blow into a less serious one.
5e's Player's Handbook has a couple bits about hit points, but the most descriptive part is on page 196.
Hit points represent a combination of physical and mental durability, the will to live, and luck.
Hit points are a reasonable abstraction by themselves, since in both the editions I know about, they effectively convey the fact that a tougher or more experienced character is better-able to survive dangerous scenarios. They also allow a novice and an epic hero to spend similar amounts of time recuperating after an adventure (since natural healing scales with the number of Hit Dice a creature has), which makes sense, given what hit points are stated to represent.
However, magical healing (be it via potions or a divine caster's spells) scales with the caster's abilities and not with the target's hit points. This means that, in both editions, an average peasant or a 1st-level fighter who drinks a healing potion will instantly heal from all their injuries and be brought back to full fighting strength. However, an epic dragon-slaying adventurer (or, in a more extreme case, an actual dragon, with its mountains of hit points) would drink the same potion, and only a very small percentage of their vitality would be restored.
What's with the difference? I know that mechanically it serves as a sink for high-level parties' gold and spell slots to force players to use stronger magical healing, but narratively, I haven't been able to find any information on why everybody's natural healing happens at similar rates, but the efficacy of magical healing is inversely proportional to a target's natural fortitude and adventuring experience. Did 1st and 2nd editions handle healing differently, or is there something specific about healing potions and magic that causes them to behave this way, or is there simply no explanation given, with the assumption being that "it's just a mechanical thing, don't think about it too hard"?