Eww. I dislike that wording. If the batter is out on the appeal, then he should be the third out...period. The defense should decide whether they want that out before they ask about the check swing.

How do they know if there has been a third out at the plate?

The PU is not going to immediately go the the BU for help, as he has the responsibility of watching for possible Batter Interference on the attempt to retire the runner.

It's situations like these that the 4th out comes into play.

[layman warning]
I got to thinking about it after I posted. Between that ruling and the conversation earlier, I had it in my head that the ruling allowed the defense to choose whether the batter would/would not come up the following inning after they asked for the check-swing appeal. Now, I don't think that's the case. If the batter is called out on the strikeout after the runner is out in this situation, the defense may choose whichever out benefits them, but the batter is still out in either case and does not lead off the following inning. Am I correct?

No. The batter is the third out. It does not matter what happens after that. The defense does not get a choice.

The "choice" in this situation is the actual appeal on the swing.

It isn't rocket science.

If there's an opportunity for a check-swing appeal then the batter could not possibly have been called out on a strikeout.

The defense has a choice as to whether or not to appeal the check-swing.

SO, as a matter of tactics:

If you want the batter to lead off the next inning you do NOT appeal. He thus cannot be out as he hasn't completed his AB so he leads off next inning.

If you do not want the batter to lead off the next inning you DO appeal. He thus MAY be out (if the call is changed to strike) and if so he has completed his AB so he does not lead off next inning.

Valid Appeal flamed
 

Well, a valid appeal for an advantageous 4th out applies to a base running mistake. A check swing appeal is simply acknowledging that the 3rd out was made before the 4th.

A 4th out is a 4th out.

It's not exclusive to a baserunning mistake. (See the FED interp previously posted)

Of course they do. Defense can let the play stand (batter is late in the lineup) and not appeal, or they can appeal (3rd or 4th batter at the plate). If they appeal with #9 at the plate that is just not smart.