Do we really need this tag, or should it be a synonym for finite-automata?
I think finite-automata is specific enough, even for questions that aim towards getting a DFA. That is because determinism is (usually) not the issue; every NFA can be algorithmically converted into a DFA. The issue is finding any finite automaton.
For questions that specifically deal with nondeterminism, we already have nondeterminism (which can be used with any machine-model tag).