That would be the answer according to a dualistic, naive conception of reality and causality (which most of us have, most of the time) but it's not the correct answer in the Buddhist version of the story.
In the traditional story, the 'third monk' is actually the teacher of the other two monks. Following their two inadequate answers, he rebukes them and says:
"It is the mind that moves."
The monks' answers are deemed inadequate because they are dualistic: they make a distinction, in a fundamental way, between the wind and the flag (and, in fact, movement as such), and then try to think whether movement begins with the one or with the other, or whether movement can be considered apart from that which moves.
But to distinguish 'movement', 'flag', or 'wind' as particularities of what is, beforehand, an unparticularised situation, is a movement of the mind. It is the monks' dualistically inclined minds which move towards a view, and any particular view is partial and therefore inadequate. So the master's answer is the 'correct' one, as it's the most accurate and apposite statement of what's happening.