You wrote that "I do not believe there is a god" means the same as "I believe there is no god". If belief equates to eating, and "there is a god" equates to an apple, then belief there is a god equates to eating an apple. Going further, "there is no god" similary equates to something that isn't an apple - or a non-apple, or other such term. So, according to you, "I do not believe there is a god" (I am not eating an apple) is the same as "I believe there is no god" (I am eating something that isn't an apple). And eating something that isn't an apple is somewhat different to merely not eating an apple - because the former implies the you are eating (just not an apple) while the latter makes no implication as to whether or not there is actual eating going on. Similarly "belief in no god" implies that you hold a belief, whereas "no belief in god" merely states that you don't hold a specific belief, but with zero logical implication as to your holding or not of the "belief in no god". Simples, really. I hope your bout of mysitification clears up soon.