Help me understand this part of the MLB save rule.

Right, you got it. Otherwise, you could say the same thing about a flyout, ground ball, or anything else where he doesn’t touch the ball.

What about a pine tar redux?

2 out and a home run is hit.
Pitcher is replaced but before he pitches the hit is appealed and the hitter is ruled out. Who gets credited with the out, the pitcher who threw the pitch or the relief pitcher?

The pitcher who threw the pitch. An appeal is credited back to the pitcher who was on the mound.

Strictly speaking, a complaint about an illegal bat isn’t an appeal play, so much as asking an umpire to correct a call. (Nowadays it would have to be for corking or nailing the bat, as the rules make clear that you don’t get called out for pine tar.)

If a player uses an illegal bat, or even carries an illegal bat into the batter’s box, he supposed to be called out at once. In reality, the umpires won’t likely notice until after the fact, if the bat breaks Sammy Sosa-style and they discover the cork. It might even be necessary for the opposing manager to point out the cork. The batter then gets called out after the fact, even if he got a hit.

But again, this isn’t an appeal play; it’s just an argument over an existing play. A comparable situation could occur on any play where a manager comes out to argue; once in a great while the umpires will confer and agree to reverse their call. Under the new rule, it can even be necessary for umpires to watch a replay before deciding to reverse.

So can a pitcher be changed in the interim? Yeah, I suppose. It wouldn’t be very smart managing to make a pitching change while an argument is still pending over a previous play, but it could happen. And theoretically, I guess it could result in the last pitcher in a game having zero innings pitched.

How much does it really matter for all these statistics baseball keeps? Is it just something people do for completeness? Or can things like “saved games”, or any other stat, determine which team goes to the playoffs in some unlikely tie-breaker rule?

No, these are for individual evaluation, not team evaluation. It can make a diffrence to a player come contract time–some have statistics-based incentive clauses built into their contract, and for others stats will come up during salary arbitration.

Forfeit, perhaps?