I think an excellent way of motivating precise definitions is to walk through a sequence of candidate weaker/less precise definitions, and for each of them construct an explicit counter example that leads to a failure of whatever property you'll often be interested in deducing in the class.

Perhaps also as a supplementary/extra credit exercise in homeworks, have similar work (albeit with some sort of mention of how if the counter example can be found on wikipedia, the explanation of why it fails must be especially rigorous), and then after each such homework is collected, walk through some of the counter examples for these questions.