So your state of mind when you realized the evidence was logically consistent would be that you still believed that "x" didn't exist,
Correct.
but you could go ahead and believe that it did exist and - poof - you would now believe that it did?
"When someone begins to believe X, then that person believes X" is what you just said. So yes, thank you for the tautology. On the other hand, a person, faced with X, can choose to reject X, and therefore, he does not believe X. (The only time this choice is irreversible is at the point of death, at which point he will have no other choice but to acknowledge X, and face the consequences of rejecting X.)
Beliefs are formed based on the acceptance or rejection of reality.
Reality, however, is what "is." It is not subjective. It is purely objective.
God exists.
That statement is, itself, a tautology, because He, being the source of everything that exists, must necessarily exist.
He is the only necessary Being.
It's like saying "the source of existence, exists."