Originally the first option only hit creatures and planeswalkers -- the flavor was that you were forcing someone to go on a quest to retrieve a card for you. When I changed it to any non-land, the flavor had to change too -- so now you're holding the first card for ransom, and the second card is payment. The "you don't control" clause was added in deference to this. I flip-flopped between the two versions, but ultimately the deciding factor was that this was the only version I could come up with a name for.

"owners' hands" or possibly even better: change it from "all cards" to "each card"/"into its owner's hand".

At first I thought the second mode was searching for the missing person/object, but it's actually preparing the ransom for the exchange. Neat.

Yeah, that sounds much better. Fixed.