Is this push-down automaton non-deterministic, as JFLAP states? There is a tool called JFLAP, which, among other things, can analyze push-down automata, and find non-determinism.
In this example it is detecting non-determinism in state q0:

The first symbol in the transition represents the symbol read as input; the second symbol represents the symbol extracted from the stack; and the third symbol is the symbol pushed to the stack. λ represents the empty symbol, so this is an empty transition without checking the stack or pushing anything to it.
I am surprised, as that state seems to fulfill the conditions for determinism for push-down automatons (if only because it only contains a single transition!). I would expect the next state to be q1 under any circumstance.
In comparison, JFLAP doesn't find any non-determinism here:

Mind you, the transition is the same, it only changes that this one adds something to the stack. Am I missing something or is JFLAP wrong in the first instance?
All deterministic push-down automatons are also nondeterministic push-down automatons, so it is correct to say that your automaton is nondeterministic.
That said you are correct in saying that your first push-down automaton is also deterministic, according to the definition on Wikipedia.