Anyway, I think this whole thread is kind of missing the root of the use of "burden of proof" when it comes to claims about god. This comes from the theists response "well, can you prove there isn't a god?" in response to atheist claims that "you can't prove there is a god." In this context, the atheists' claim that the burden of proof is on theists is correct, as the theists are the ones making a positive claim (that god exists). Unless, of course, you're talking about atheists who claim that god doesn't exist. Then they would also have a burden of proof for their positive claim. Regardless, I think this thread isn't really connected to the core of this issue, and how/why the "burden of proof" argument came up.
You stopped before the part that this thread addresses.
So we have this:
Atheist: You can't prove that god exists!
Theist: Prove that god doesn't exist!
Atheist: You have the burden of proof!
This is the argument you described in your post. But, the debate doesn't just end there. My question is "What about what comes next?"
Theist: God exists because argument X, reason Y, and evidence Z.
(This is the theist's attempt at fulfilling her burden of proof.)
Atheist: You have not convinced me, therefore, you have failed to meet the burden of proof.
(This is the atheist rejecting the proof offered by the theist.)
I am not arguing about the part of the debate you described in your post. I am talking about the next step. The theist has offered proof and the atheist has rejected it.
Does the atheist need any reason to reject the arguments offered by the theists?
Is she allowed to reject them "just because"?