You said that he is absent the belief that at least one god exists. You then said he does not believe that gods e,f,g do not exist. So, if he has a conception of god, but cannot evaluate that conception because of some lack of knowledge then he is an agnostic. You are saying that Jack has a conception of god. You are saying that Jack is absent belief in gods existence or non-existence existence, despite having a belief in the non existence of specific gods. We call this agnostic because to jack, cannot evaluate the proposition because he lacks some form of knowledge to evaluate the proposition but recognizes the proposition exists. To him the proposition is unknown or unknowable. You are needlessly contemplating things.
Contrast jack with myself, even without knowing what gods e,f and G are, I believe they do not exist as gods. Either they simply have no basis in reality or they exist in reality but are not gods. Because I can evaluate the proposition I am not an agnostic. I am simply an atheist. Now the term agnostic has also been distorted. So if you were to press an ask whether or not I know, that no god exists I would say no, I do not claim 100% certainty. That is: I lay no claim that my belief is a justified, true, belief. However, if we are talking about belief without rising to the level of a justified true belief then I make the claim no god exists.