2 strike 2 outs

[Altor]

[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?