Your hit points represent a combination of several factors. They include your physical durability and overall health, your speed and agility to avoid harm, and your overall level of energy. They also account for luck, divine favor, and other mystic factors.
In short, hit points are an abstraction. While you are at or above half your maximum hit points, you show no signs of injury. At less than half your hit points, you have acquired a few cuts and bruises. An attack that reduces you to 0 hit points or fewer strikes you directly, leaving a bleeding injury or other trauma, or it simply knocks you unconscious.
The 5E description of HP is treating them as additive, in discrete chunks: luck/fatigue on top of cosmetic damage on top of actual injury. It makes sense that you could "second wind" to restore some of the first chunk, but the second and third chunks would require magical healing or just take forever. A lot of the issue that I've been having is when you're very low on HP, because you shouldn't be able to "shrug off" the actual damage portion, and it feels goofy to use divine blessings to make you feel slightly less tired. This system could actually work really well if everyone had temporary hit points that you could "second wind" through and real physical hit points that you couldn't, but it might require too much bookkeeping for casual use.
I think it could solve a lot of problems if they approached HP as multiplicative: your actual structural damage capacity is spread out over your HP, and then luck/fatigue/training represents a reduction in the amount of injury caused by an attack. It's like, a first level character has 10 HP and a tenth level character has 100 HP. A dagger wound at 1st level causes 5 damage because it's all actual damage, but a level 10 character can use skill/luck/energy to reduce that down to a tenth of the effectiveness - a minor scratch. Every attack that does HP damage represents an actual wound, but most wounds are minor. (I actually borrowed this interpretation from the armor system in the old Synnibarr. Check them out on Kickstarter.)
Of course, the problem with that model is that it becomes harder to fix a minor scratch in an accomplished hero (10 HP) than a minor scratch on a rookie (1 HP). That's why I favor the "normally I would be dead now" model. Every wound is equally bad for everyone, and our intrepid heroes are just better at not dying from it. Some guy might keel over after you stick a dagger in his gut, but Brock Samson is just going to ignore that and snap your neck. It still takes 5 points of healing to cover the wound, because it's still the same wound. Whether it's divine luck, endurance training, or however you want to explain it, you're not dead after being shot with a dozen arrows and set on fire. Afterward, though, you're going to need a month in a hospital or (preferably) divine magic. This method also has the benefit that everything is exactly what it says it is: a hit is a hit, a critical hit is a really solid hit, healing is actually healing, etc. (The first method is much harder to narrate: "He swings his axe and comes very close to hitting you before you narrowly dodge - take 8 damage.")
The metagame is not the game.
This started in a different thread, but I didn't want to derail it too badly so I moved it here.