Does it make sense to define atheism as "a lack of belief in a God" rather than as "a belief in the nonexistence of God?"

Every atheist lacks belief in a God; but lacking belief in a God is not sufficient to make one an atheist. Consider babies, squirrels, or stones. They lack this belief (and perhaps all beliefs) but it seems rather odd to describe them as atheists. We presumably want to reserve this term for those who have considered the arguments and evidence on both sides of the issue.

If we can neither prove nor disprove the existence of a 'God', is it rational to even consider the possibility that he/she exists? Without the dedication of the few who preach from the worlds' religious houses, the notion of a 'God' surely wouldn't cross the mind of even the most imaginative of thinkers?

You ask two distinct questions. The first is whether it is rational to assert that God is possible, assuming that you cannot prove or disprove that God exists. The second question is whether the concept of God would occur to anyone who is not influenced by the preaching of religious teachers. About the first question: when it comes to questions about "possibility", it always useful to distinguish various senses of this term. "Logical" possibility depends upon whether a statement involves a contradiction or not. So for example, a square circle is impossible. But a piece of gold that weights a trillion tons is not. "Epistemic" possibility is significantly weaker: it refers to whether or not a proposition is possible "as far as we know", or relative to our knowledge. It would seem that the statement "God exists" is possible on either interpretation. It does not appear that the concept of God contains any contradictions. And if you cannot disprove that God exists, as you stipulate, then God is...