How do you handle hit points?

I say something like “the orc hits you for 19 points of damage narrowly missing decapitating you”.

I consider hit points and damage this way:

Someone hits a Fighter with a sword, doing 8 points of damage.
Someone hits a Wizard with a sword, doing 8 points of damage.

Why is one more injured than the other, if they're the same level?

The fighter, having been trained in melee combat, has learned to turn the blow, shifting in an almost instinctual manner so the blade doesn't some much dig deep but instead produces a long shallow cut.

The Wizard, who didn't spend days on the practice field beating on, and getting beaten on by, trained instructors and other trainees, can't "roll with the blow" nearly as well.

So both blows do the same number of hit points, but the Fighter knows how to take the hit better, which accounts for his higher hit point total over all.

I definitely fall into the camp that every successful hit makes contact (breaks through the defenses and actually leaves a mark on the body of the opponent.) This is to account for things like poison and necrotic damage etc. If the blow was actually avoided then there's not much chance of poison taking its toll. So HP is your ability to field the blows, and stay in the fight. It's pretty much the same for monsters, you're whittling them down to a point where they make a fatal mistake and a killing blow is able to be landed. I generally narrate the PCs blows to be more damaging than what they receive, because that's more fun and epic.

This concept (not new, I know) of hp = fighting spirit also makes it easy to realistically handling killing etc out of combat. If you want to kill a monster/NPC that is not in a position to fight back then it just happens (well as long as the PC knows the vulnerable spot of course, so that might require a check ). But no attack roll is needed. Same with knocking one out (with some reasonable restrictions).

HP is tracking the "fight" in an opponent and all successful hits make contact with flesh however superficial.

For the sake of consistency, and everyone at the table having a clear idea of what the game mechanics actually mean, I narrate damage the same regardless of the HP capacity of the victim. If you take 8 damage, then that's a solid (but non-critical) strike from a sword, being wielded by someone with above-average Strength. If you take 16 damage, then you're hurt twice as badly as that.

The main difference between heroes and chumps is that a hero doesn't give up the fight, just because they catch an arrow in the shoulder. Ask yourself how many arrows Batman could take, before he gives up. So I go ahead an narrate him taking arrow, after arrow, after arrow, and not dying. You'd think he would die, because most people die when they take an arrow or two, but Batman is amazing because he never gives up.

As a side note, I've extensively re-written the healing rules to get rid of Hit Dice and to slow down the rate of recovery through sleep, because there's no way to reconcile damage such that it is both potentially fatal and leaves absolutely no evidence after you take a nap.

HP are a combination of luck, stamina and physical toughness. Damage is a combination of nicks, bruises, strains, pulled muscles, cuts and exhaustion. That last hit that drops you does more damage because you've been beaten up enough that you can't quite get out of the way or deflect the blow quite as well.

As far as how quickly people heal, I use the alternate rule where a short rest is overnight and a long rest is several days, usually a week or more. This also makes HP and damage a touch more realistic because you can put up with quite a bit but over time if you keep getting beat up every day you just don't have any reserves left.

Of course D&D isn't particularly realistic in the first place, it's more of an action movie reality.

The other players narrate the damage to their character. I have them roll the damage the monsters do to themselves and a player may say "I take 8 points, a minor wound", or "Help, cleric..."

For monster damage I may do some general things and sayings. When a big monster gets to about half HP, I may say they are starting to look bloodied, and when they have only a few HP left I place a red mark on them. This is a hint to PCs with 2 attacks that only 1 attack may kill the monster and he could move to another for the 2nd attack. I also like to throw lots of little monsters at higher level PCs and may just tell them that the skeletons have 13hp. I also have 1 hit and 2 hit monsters in some fights where I will just put the red circle on each monster after one hit or take away the monster if they are little. Some may have ended up with one or two HP left, but the hand-waive works in some situations.

I generally describe the monster's damage based on their HP total. If the attack leaves the monster with more than 50% of their HP, the attack made contact, but had little visual effect. If it puts the monster between 50%-25%, I give it a good physical description of a solid wound. If it puts it less than that, I describe the pain (and usually fear) the creature is in, with blood flowing freely from its wound(s). Kill shots often get a simple, but obvious description.

With players, it's a bit different, since I don't track PC HP anymore. If I know the character is still at high health, I'll describe the impact of the blow, with maybe a superficial wound. If I know the character is low on HP (usually because they've said so), then I describe it as a deep and painful wound. "Kill" shots get a more detailed description, but since PCs can come back, I always make them less gruesome.

First off, are you sure you don't want to switch to GURPS... One good sword blow can kill anyone no matter the experience. That system has Parry, Dodge and Block options to evade the blow or take the hit on a shield. If your players find themselves wanting to put more details into their combat, like "I want to hit the orc in the head" then they want to play GURPS. Steve Jackson Games. A lot more detail, but a lot more realistic combat.

As far as description of the melee I reserve details for huge hits. For instance the Rouge hits an Orc with her thrown dagger and does 5 points of damage. No need for description. The Ranger fires his arrow from his +1 longbow using hunters mark and all the other stuff he gets at 5th level and does 39 points of damage to the orc. That deserves a "The orcs heart is ripped from his body and stuck to the building behind him." The Paladin rolls a Critical Hit on a Smite attack with his magic sword and deals 30 points to the Dragon. That deserves a "A eight inch long slash is ripped open across the right forearm of the dragon making it recoil in a pain it hasn't felt in a century." If the Veteran Fighter gets smacked for 1/3 of it's Hit Points a "He steps back shaking off the pain of your strike. His next swing seems less powerful than the last one."

I avoid talking about specific body parts being severed. A wound to a body part is fine as long as you remember to not say it again or else some player is bound to say "is it cut off yet?" slowing everyone down. If the player is obnoxious enough remind them that whatever they do to the badguys (like target limbs and such) you will have bad guys do to them. Makes that "I used to be an Adventurer until I took a Warhammer to the knee" meme take on a whole new meaning.

The Joy of D&D is the ease of getting through a combat. With eight players I can get through four heavy combats in four hours with D&D. With four players I can get through two similarly heavy combats in four hours with GURPS. SOme nights of GURPS I plan the whole session as a combat with some groups. Not joking. Threaten them with GURPS a few times and they will be happy for D&D.

I subscribe to the idea that "HP" is an acronym for cHutzPah.

I run hit points as a combination of meat, skill, luck, etc. When describing hit point loss, I roughly follow the 5e outline. About the only difference is that if I need to know how many hit points are meat, I use con score. If a PC has a 14 Con and 58 hit points, 14 of those are meat.