Ya, it's not really up to atheists to define to Christians, Muslims or whomever what their religion is about. It's their God, so they're the ones who need to define it.
I've been at pains to explain why this isn't true from a purely linguistic perspective, but you aren't the only one to reject the idea. The problem is that "define" has at least two senses. In one sense, it means "to provide a definition of common usage". In the other, it means "to prescribe how one ought to use a word".
Let's take the first sense--providing a definition of common usage. That is a straightforward empirical issue. The first thing you discover when you look at the word "god" is that atheists, Christians, Muslims, and people of whatever religious belief all use the
same word sense for the common noun "god". Its meaning isn't tied to the religious identity of the speaker, because the common noun does not refer to the specific deity or deities that the speaker believes in. People of all religious beliefs also name their specific gods. Christians use the proper noun "God" (note the capitalization) to refer to theirs, and Muslims tend to use the Arabic derivative "Allah". The labels "theist" and "atheist" do depend on the meaning of "god", but not "God". Someone can believe that God does not exist but still believe in deities of other sorts. Such a person would still be a theist. And if someone disbelieved in all deities but the Christian God, that person would still be a theist. So your generalization is not right with respect to the first sense of "define".
You could have been talking just about the second sense--where speakers of the individual religion prescribe the intended usage. The thing to remember here is that they are not prescribing usage of the common noun. What they tell you about their specific gods have to do with concepts associated with just those particular deities. Since Christians tend to have a very strong taboo against belief in other gods, they will only try to define the one they believe in. But that would be a definition for "God", not "god". Those are two different words.
Tom, I want you to look at the spelling in your second sentence. You wrote "their
God". Technically, that's a punctuation error, because you were capitalizing the common noun, not the proper noun. If you intended to use the proper noun, it would have been ungrammatical to use the so-called demonstrative adjective "their" as a modifier. Proper nouns don't tend to allow modifiers in English syntax. I think that the slip was the result of equivocation on the two senses of "G/god". People of specific religions have every right to prescribe common usage for specific gods. It's just that atheists do not reject belief in particular gods. They reject belief in all gods. So much of what you get out of Christians isn't going to be helpful in defining either theism or atheism, just a particular instance of god-belief. Maybe that is all you want for the purposes of discussion, but that is going to draw you into a lot of minutiae that have little or nothing to do with theism and atheism generally.