Third Out on Steal -- Who Bats Next Inning?


 
There are two outs, with a runner on second base (R2), and a 3-2 count on the batter (B1). The pitch is in the dirt, so B1 does not swing. The pitch ricochets off of the catcher's shin guard and rolls up the 3rd base foul line. The catcher sees that R2 is now trying to steal third, so he throws to third base and retires R2 for the third out of the inning. When this team returns to bat in the next inning, who leads off the inning -- B1 or B2?
 
Bonus Question: In the scenario above, what if B1 swings and missed at that pitch in the dirt? Since this is an uncaught third strike, B1 is not yet retired. Since the catcher chased the ball up the third base foul line (as in the play above) the catcher elected not to make a play on B1 but threw to 3rd base and retired R2 for the third out of the inning. Now, when this team returns to bat in the next inning, who leads off the inning -- B1 or B2?
 
Call
A batter's time at-bat ends when he becomes a batter-runner. When his time at-bat ends, then the player who follows him in the line-up is expected to bat next.

In both situations below (a walk and a strike out), the batter's time at-bat has ended. To start the next inning, B2 is the proper batter and B2 should lead-off the inning.