Hit points in any edition of D&D are far too abstract to actually say that losing hit points counts as "being injured" in any physical sense, save for three specific contexts:
(1) The minimal amount of hp possessed by a low-level/low-CR/low-HD creature (depending on edition);
(2) A certain proportion of hp possessed by a very large creature (that has a lot of "meat" to cut through to inflict a mortal wound);
(3) The last few hp of a PC/NPC/monster that has a lot of them.
As such, this reading of AC doesn't make sense by my reckoning beyond those contexts.
In fact, given what hit points have varyingly been said to represent across the editions, "damage on a miss" is
entirely sensible within the fiction.
And indeed, at least one OSR game - Worlds Without Number - has a mechanic that is very close to "damage on a miss" as implemented in 4e, in the form of Shock (
free edition pg. 43), which to me suggests that "damage on a miss" feeling incongruous to a player is not the result of some genuine fact about what is going on in the fiction, but instead the result of how that player conceives of the goings-on in the fiction - which is fine as far as it goes, but it also means (a) that
they could just conceive of those goings-on in a way that reconciles with the mechanics instead of in a way that doesn't; and (b)
if they do not that is a choice they have made. (It's their choice to make, sure, but it still is what it is.)