I take it that this last entry of yours addressed your earlier line: "A set is denumerable (contains a denumerable amount of members) if its members can be mapped by some rule or rules, one to one, with the positive integers, and that set can contain either a finite or infinite amount of members." Making only the first change you propose, we get: "A set is countable (contains a countable amount of members) if its members can be mapped by some rule or rules, one to one, with the positive integers, and that set can contain either a finite or infinite amount of members." One more refinement of this is needed: "A set is countable (contains a countable amount of members) if its members can be mapped by some rule or rules, one to one, with some or all of the positive integers, and that set can contain either a finite or infinite amount of members."
The proposal in your second paragraph is right, though it would profit from a little smoothing out: for the last phrase, put "and that set will contain an infinite amount of members".