But, if someone doesn't want to admit that logic exists or you just disagree with someone as to what logic is, there's really nothing to be done but to walk away.

That's not necessarily true. If we disagree on what logic is, I can work out the rules of what you consider logic and decide whether, using those rules, I come to a different conclusion than you do (in which case I can try to convince you of that different conclusion using your rules), or I can attempt to convince you that you're wrong via illogical means (like telling you a convincing story, or using question-begging language, or etc.). I can also do the latter if you reject logic altogether.