That's similar to how I look at it. I basically think of it as a retcon: if I really got shot, I obviously wouldn't still be going, so it must just have been a near miss that freaked me out. The ones that kill me/incapacitate me are the only "real" hits.
This. So much this. And also the game saving you in an impossible situation because you simply do not have enough health to do what needs to be done (mostly only an issue when there are infinite enemies).
Yeah, that's how Halo: Reach does it too, and I like it a lot. Your health is divided into 3 blocks, if you lose a whole block it's gone until you pick up a health pack, but if you only lose part of a block then that block will regen. That's of course in addition to the shields, which regen completely, though in a game where shields don't make sense, like CoD or whatever, I'd still be perfectly happy with the block system.
Basically it means that as long as you don't actually die, you can regen at least up to 1/3 of your total health, and that's enough to give you something to work with.