stormlight

Thank you !! By the rule does the runner on first need to tag or wait for the ump to call the out before they can run to second?

Hello all, Often times at youth league games i see this scenario and I think it is wrong. Can someone clarify this for me. Runner at first 0-1 outs. Batter at the plate Drop third strike happens. Batter starts to run to first base or hard fakes to first (Because they know the rule that they are out) Catcher throws to first (because they don't know the rule or are caught in the moment) and then makes a bad throw to first. Runner that was at first takes off to two Batter is called out by ump but also stops the play and calls runner back to first. My understanding is all that should happen is that the ump should make the out call at home plate. He should not stop the batter from running to 1st and not call back the runner from 1st to 2nd as a drop 3rd with less then two outs is not a dead ball play. It's simply just an out for the batter and a live ball. Am I correct in my understanding and are there any rules that prevent the outed batter from running to first or hard charging to first but stopping half way to try to get the catcher to make an errant throw to first? Thank you!!