3rd strike checked swing, uncaught by catcher
 

USA As it was told to me: No outs, no base runners. Two balls, two strikes on batter. Batter checks her swing, ball goes past catcher to the back stop, batter runs to first base. No call by the PU, who does not ask FU for assistance on the swing. Defensive coach asks if the batter swung for a third strike. This does not appear to be a play that can be appealed, according to my understanding of appeal plays. However, I am seeking opinions.

Seems to me if the PU let the runner stay at first it was called a strike so the "no call" claim has to be incorrect.

In baseball at least you cannot appeal a strike call.

PU should signal a strike if it was a strike. There is no USA signal for "the pitch is not caught." Why does the defense want a swing if B is standing on 1st base? He wants a ball.

Maybe it was called but everyone was too busy watching the catcher chase the WP. Batter knew to run after all.

It is not what comes under the heading of an official appeal. But many times the word "appeal" is used when the defense wants the PU to check with a partner (as long as the partner is in a position to help).

PU should have verbalized a ball or signaled a strike if s/he believed the batter actually swung.

Simply asking the partner for help on the checked swing (which some folks use the "appeal" word) could have helped.

But if it was a ball, obviously bring the batter back to complete the time at bat with a 3-2 count. Batter ran thinking it might have been a swing.

It is actually called an appeal in baseball rules.

As you say, asking the umpire to check with a partner on a swing is not an appeal in softball and arguing balls and strikes is not allowed, however, in this case, I'd be doing neither. I'd be saying, "Blue, you didn't make a call on that pitch, wasn't it a ball?" [the OP said the umpire made no call. that's what i would be questioning.]