Proof that PDA's with different definitions have same expressive power
You prove it the same way you prove that any two models of computation are equivalent: you show that, for every machine of type&nbsp;A, the language accepted by that machine is accepted by some machine of type&nbsp;B, and vice-versa. Sometimes, this is trivial, for example if A is just a restricted version of&nbsp;B (e.g., proving that every language accepted by a DFA is accepted by an NFA). In other cases, it involves simulating the features or acceptance criteria of&nbsp;A using&nbsp;B (e.g., proving that every language accepted by an NFA is accepted by a DFA by using the subset construction).

If you're looking for details of how to prove all the equivalences mentioned in your question, then I think your question is too broad. You should be able to find some of them in any good textbook that covers PDAs, and you'll probably find the rest as exercises because the specific proofs probably use the same general principles.