"An Atheist does not believe there is not a higher power" is not a double negative. Its basic logic that I can't really break down further.
You should say "Christians believe a higher power exists.
Atheists do not believe a higher power exists."
That was a type I ment to say "I don't believe either until you present evidence."
Those are opposites. You are mixing up a single position on one claim with, a positions on two claims. "There is a god," is a claim that I don't believe. That makes me an atheist. A person who does believe is a theist. "There is no god," is another claim. I don't believe that either. There is no word for that because its an impossible claim to prove. You can't prove there are no gods in the universe. You haven't looked everywhere in the universe. Theists can't even agree on a definition for what a god would be.