Sheepishly, another backcourt question

[imagomer]

Please correct my understanding as I try to back up a no-backcourt-call with rules. A1 is dribbling in his front court. Defender B1 bats the ball off A1, then into A1’s backcourt. A1 is first to touch the ball in a1’s backcourt.

I think this is NOT a violation because team control ended when the dribble ended (4-12.1 and .2a), and the dribble ended when the defender touched the ball and caused the dribbler to lose control (4-15.4d). Is this sound reasoning? If so, then please tell me why 4-12.4 does not seem to apply. (4-12-4: while the ball remains live, a loose ball always remains in control of the team whose player last had control, unless it is a try or tap)

Would the ruling change if A1 had accidentally dribbled off B1’s foot, ball touched player from A, ball into backcourt, and A1 is first to touch?