I agree with the second suggestion, but not the first one. You're certainly not going to feel refreshed when waking up after being knocked out by a combination of injury and fatigue, but you're going to recover from some of the oxygen deficiency and some of the lactic acid effects.
I'm not opposed to the introduction of specific effects of being knocked out by blows to the head, but that would be separate from the current encumbrance+injury+fatigue effect (which kicks in regardless of whether the fainting was done as a result of being hit on the head or bitten in a hand [yes, I had a character that died due to the latter: failed 3 75% effective chance actions (thrown javelin, javelin thrust, dodge), got bitten in the hand, fainted from pain, and the wolf made sure the character never woke up again]).