Two instances:
1) Player controls ball on side, rolls onto back.
2) Player controls ball on back, rolls side to side in order to release pass and/or avoid tie up.
When would traveling be called in each case?

I would kind of need to see the player to see exactly how much rolling they are doing.

Based on your description I would probably no call that. I would view him as still on his back until they flipped all the way over to the front.

4.43.5 SITUATION B: A1 dives for a loose ball and slides after gaining control. A1 is in a position either on his/her back or stomach. What can A1 do without violating? RULING: A1 may pass, shoot, start a dribble or call a time-out. Once A1 has the ball and is no longer sliding, he/she may not roll over. If flat on his/her back, A1 may sit up without violating. Any attempt to get to the feet is traveling unless A1 is dribbling. It is also traveling if A1 puts the ball on the floor, then rises and is first to touch the ball.

A partial roll? Probably not. But if he was rolling to avoid losing the ball, easy, traveling.