1. "Proof" and "Evidence" are often synonymous.
No, they're not synonymous. Proof provides certainty relative to a set of axioms. Evidence may support a model, but no accumulation of evidence can provide certainty. Science is a realm in which there is no certainty, regardless of the strength of the evidence.
2.'Logic' isn't used in science or philosophy?
The validity of a proof is only as good as the validity of its premisses. Neither science nor philosophy has been well founded, systematized, and formalized sufficiently to enable the use of logic to provide persuasive proofs. I doubt that they ever will be.
Logic is used as a tool in science and philosophy. The result of using logic in these domains is suggestive rather than conclusive.
So how are you different from the creationist who says "I don't think I could be persuaded that Evolution occurs"? He is close minded and has no justification for denying Evolution if he can't even make it clear what would convince him of Evolution.
I have great respect for the concept of proof, where it is relevant, and where it is not. I've already said that I don't think that 'proof' has any meaning in science or philosophy (including religion).
You asked what I would accept as PROOF of the existence of a god.
There can be no proof of the existence of a god.
There can be no proof of evolution.
There is an extraordinary amount of strong and persuasive evidence for evolution.
There is no strong or persuasive evidence that a god exists.
There are things that could persuade me that there is a god, though I have a very high threshold for persuasion these days. I've been religious in the past. I am an atheist now. I have no idea what will happen in the future.
I know that my thinking is flawed, that I am ignorant of most things, that I can be tricked, that I can deceive myself, and that I am often irrational.
I share these shortcomings with everyone.
I also know that I and many others take medication for approximating mental health.
There is no certainty in mathematics; there certainly is none in the rest of the world.
If the only evidence for the existence of a god is a phenomenon that I experience, I suggest that it is far more likely that my perceptions and thinking are flawed than that there is a god.