You ask how I can tell someone that I don't believe in god without defining the term -- whose god don't I believe in, mine or theirs? That's a good question. From my perspective, since I say that I don't believe in god since I have seen no compelling proof for the existence of a deity, I feel that I'm covering all possible definitions of god. It is a true statement that I have not seen any compelling proof for anything that I could meaningfully label 'god.'

You seem to be saying that all possible definitions of "god" are limited to what you could meaningfully label "god". So, if I decide to use "god" as synonymous with "rabbit", you would say that it is an inappropriate label. Clearly, you must have some minimum requirements that an entity ought to fulfill before it can deserve the label "god". I would very much like to hear what you consider those to be.

I say that the inability to define something does not prove that something does not exist

Maybe, but the implication of this statement is that you believe we can still apply the predicate "exists" to an entity whose existence you agree we cannot define, I find this absurd.

From a philosophical perspective, I would say that a minimum definition of "god" for the purpose of this discussion would be something that is not bound by the laws of nature, is not confined to our universe, and has or had the ability to interact with our universe. This definition doesn't cover everything that some people call god ("god is nature," "god is love"), but I consider those types of definition to be in a separate class, one that doesn't impact whether I call myself an atheist. We might be able to come up with a better definition if we wanted to work on it a bit.

For practical purposes, though, I'd say that "god" is any supernatural thing that many people would label as a deity if it existed. That's a much easier definition to work with, and it again puts the onus on the believer (where I think it belongs). It also gives some pretty lame things the opportunity to be called god, but I can live with that.

You talk about the inability to use the term "exist" in relation to something undefined. You are right and I should have chosen my words better. Perhaps I should have said that we cannot make definite statements about things completely outside of our experience. I would also say we can't say something doesn't exist just because we are unable to investigate it scientifically.

Another point, because I think I let our definitions drift a little -- I wouldn't say that we can't define existence outside our universe, but rather that we can't describe it. So you could counter this if you could show that "existence outside our universe" is meaningless in the same way that "temperature below absolute zero" is meaningless.