[MD Longhorn]

None of those three illustrate the relevant point. I think I agree with you (mostly) on all three of those that you just posted.

1) Appeal denied - B3 is the correct batter.
2) If appeal is honored, the only penalty here would be putting B2 in the box. So see below.
3) Appeal denied - B3 was the correct batter.

Here's the one I think we disagree on (pulling aside any fluff).

B2 bats for B1 and singles. B1 then bats advancing B2 to third. B2 scores on a wild pitch. B3 hits a single, scoring B1. THEN the appeal is asked for.

7-2-D-4 says that if AT THE TIME OF APPEAL, the proper batter was on base, that batter is skipped. But in this scenario, B2 was NOT on base when B3 singled. B1 was the previous batter.

The added wrinkle is this - given that there was nothing to alert umpires to any need to memorize who was on base at any point ... it's possible (maybe probable) that the umpire is unaware that B2 was on base way back at the beginning of B3's at bat. All he knows is that B3 just hit, and he might know that B1 just crossed the plate. Asking the scorekeeper (unless we've got a collegel level scorekeeper and not just some parent) MIGHT get us the information that B1 was the kid that just scored on B3's hit, if PU didn't happen to notice and/or everyone was already in the dugout when this appeal is made.