In regards to the NCAA, the only place I've seen language about an airborne receiver having to maintain control, or "going to the ground on their own" is in the ARs, not in the rules.
That suggests (to me anyway) that those are not rules, but rulings - which means that there isn't a RULE that says that a receiver going to the ground is any different whether he is going on his own, or as a result of the defender. The ARs are not exhaustive, so the fact that one includes language like "...an airborne receiver must maintain control of the ball while going to the ground..." does NOT mean that a non-airborne receiver does NOT need to maintain control as well.
The ARs are kind of maddening. They have one example of an airborne receiver losing control when he hits the ground, and state that it is not a catch, but don't say anything about a non-airborne receiver doing the same thing - is that because it is different, or because the AR is attempting to make it clear that an airborne receiver has to maintain control just like a non-airborne receiver?
Same thing with "on his own" AR 7-3-6-XII uses this language: "Any receiver going to the ground on his own in the process of maintaining a catch must maintain control of the ball when he hits the ground". Ok, great. But does that mean that if he is getting tackled and doesn't maintain control, ie NOT on his own, then that is a completed pass? Surely not!
I think this AR is to illustrate that even if the player is all by himself, he STILL must meet the same standards as if he was NOT all by himself - not that the standards are different.
From a rule perspective, there is silence. Rule 2-4-4 defines a catch/posession, and rule 7-3-6 a completed pass, and neither pf them say anything about a difference between an airborne player and a non-airborne player as it relates to controlling a ball while going to the ground assisted or otherwise. You could look at 2-4-3-1 as it relates to 2-4-1 to conclude that a player not in flight does not need to control it to the ground, but AR 7-3-6-XII makes it clear that at least when he is going by himself he certainly does. I can't imagine that a player being tackled is not required to control the catch through contact with the ground as well.