I agree with what Sarkus wrote in post #630.
People, whether theists or atheists, seem to advocate for the truth of propositions.
When he or she is making existence claims, the theist is advocating for the truth of the proposition 'God exists'. From the point of view of the atheist, the proposition 'God exists' is false. So from the point of view of the atheist, the theist is advocating for the truth of a false proposition.
Atheists appear to advocate for the truth of the proposition '~(God exists)', or 'God doesn't exist'. From the point of view of the theist, '~(God exists)' is false, so the atheist would seem to be advocating for a false proposition too.
I don't see any big logical difference. Both sides are advocating for the truth of propositions that their opponents think are false.