Most people would say 'hang on' to tell someone to wait and not leave, and 'hold on' to wait on the phone, though the phrases could be used interchangably.

either one works just fine :)

Wait a second means wait a moment. I would rather say "Wait a moment" because some moments are long and some are short. Also, Wait a second has nothing to do with leaving. It's about waiting. "Hold up" means hold your position... stop walking... stop... stand there... wait for me. "Hold on" can mean "wait for"... as in wait for me to finish what I am doing...or anyone else that is doing something and not able to give you attention at that moment. "Hang on" is meant to have the other person wait... just wait... for you for whatever reason. It doesn't imply that you are leaving either.