Deterministic vs. Non-Deterministic PDA?
Your definition is wrong. A PDA is non-deterministic if in some state there are several possible transitions. It doesn't matter if that applies to a transition to a final state.

Your example is deterministic. Check:


From state $q_0$ with $Z_0$ on the stack, on reading $a$ there is one possibility. In the same case there is no alternative on input $\epsilon$
Similarly, from $q_0$ with stack $a$, only one option. No $\epsilon$ alternative
In $q_0$, with $a$ on the stack and inputs $a$ and $b$, again one move only. No $\epsilon$ alternative


You can complete the analysis for the other possible combinations of state, stack symbol, an possible input, making sure there is one option for a given input symbol, and if $\epsilon$ input is allowed, no "real" symbol alternative is available (e.g. in $q_1$ with $Z_0$ on the stack, only $\epsilon$ can be read).