It's implied in my last post, but let me state explicitly that a batter's status is also declared retroactively when he hits a ball that is uncaught. When he hits the ball, he apparently becomes a B/R. If the ball becomes fair, his status is confirmed as a B/R, and whatever he did/does as B/R "happened". If the ball becomes foul, his status reverts to batter, and in effect, he never was a B/R.
Excepting bunts with two strikes, of course!
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