Naturalism more fundamental than atheism?

Occasionally a theist will inquire, "What evidence would convince you of God's existence?" There are a few stock replies to this, one that I see frequently is: The moment a phenomenon manifests in the natural world in an observable way, it ceases to be supernatural. In effect, this means that actually showing evidence of a God is evidence that the God is not supernatural. In short, evidence for God is evidence against God.

Now, this strikes me as a cheap shot, though I do agree with it. It seems like a conversation stopper. But what this leads me to ponder about is what my core beliefs are. And it makes me think that naturalism is fundamental and atheism flows from naturalism. I shouldn't call myself an atheist. I should call myself a naturalist. The position of naturalism not only dismisses god claims, it is a tool for dismissing all supernatural claims. I am an asupernaturalist more than I am an atheist. Lacking a belief in gods is merely a subset of a more fundamental world view. This worldview is opposed to supernaturalism and champions science as our best tool for distinguishing noise from data, myth from reality.