Strictly speaking I do not answer your question, but I will hint at a broader automaton model. (It is meant as a PS to the answer by Yuval.) I will not anwer questions here: it might get off-topic.
The nondeterministic automaton is "existential": it accepts a string iff a run on that string exists. How you want to visualize that is your own choice, and Yuval has sketched three possibilities: try all possibilities in parallel, one-by-one, or accept divine hints.
There is a larger type of automata, called alternating, that have existential and universal states. In an existential state one requires the existence of at least one succesful continuation, while in a universal state all continuations should end in a final state.
For finite state automata, the existential extension is equivalent in power as the original FSA, whether deterministic or nondeterministic). For Turing machines there is an additional layer of complexity classes for alternation: e.g., $P \subseteq NP \subseteq AP$. And there is an even much nicer connection to games and logic: alternation is like a two player game.