When does a PDA split?
A PDA (or an NFA) doesn't split. At any given point in time, it non-deterministically chooses a valid next step, if any. If there are several options, you can say that the PDA "splits" if you wish, trying all of them in parallel.

A PDA (or an NFA) could have more than one possible move even if it has no $\epsilon$ moves, and conversely, there might be situations in which an $\epsilon$ move is the only possible move of the automaton (can you think of such a situation?). So your identifying $\epsilon$ transitions with "splitting" is wrong.