Clearly, if the hit points aren't at 0 or lower, it didn't hit a joint or sever a tendon, yet still went deep.
Meh, if the arrow 'goes deep,' but courteously evades, muscle, bone, tendon, ligament, vein & nerve, so as to avoid inflicting any temporary impairment, permanent disability, or progressive damage, is that really any more plausible or 'serious' a wound than a scratch, bruise through armor, or Gygaxian pseudo-hit? No. You've got a nominal wound that has no effect but to degrade your plot armor, and there's no reason you can't get that plot armor back in the next scene by whatever means fits the story (a short rest, a little amateur first aid from the attractive victim you're rescuing, some genre-atypical glowy healing from a pious buddy, an experience, conversation or memory that re-iterates your commitment to the mission, or whatever).
The exact same blow is ambiguous, isn't it? If we look at it from the attacker's point of view, the exact same effort and skill against a 15th level character is parried, or dodged, or somewhat deflected; or just as the attacker is about to connect s/he slips slightly in the mud and the blow is merely glancing (that would be one way the defender's "luck and divine protection" might manifest); etc. Hence, the 15th level character is not badly hurt.
If we look at it from a perspective that includes both attack and defender, a 20 hp hit on a 15th level PC is not the exact same blow, precisely because the high level PC is luckier and more skilled.
So it is the 'exact same blow,' from the attacker's side, just with a different effect. I mean, the 'exact same blow' that decapitated an ogre will miss a Halfling, right?
Seriously, though, the rationalization EGG had in the 1e DMG was that hp damage that would run a normal person through only scratched the more experienced, higher-hp, fighter, due to skill, luck, 'sixth sense,' &c. So, same blow, same hp damage, completely different wound, same level of Cure Wounds spell required to get those hps back. While that rationalization explained the seeming absurdity of 70 hp humans riding 14 HP warhorses, it created it's own absurdity of non-proportional healing.