I don’t know. In the end, there’s a single reality. Logic isn’t some kind of independently existing entity–it’s a method of reasoning which attempts to conform with reality. Hence, there ultimately can only be one true conception of logic. If different conceptions of logic conflict, they can’t both be correct, any more than we can be living in different realities. Something has to give.
As an (important) aside, our knowledge of reality has limits. Therefore, our understanding of logic has limits, and logic may not be directly applicable in all situations (for example, what is infinity? Can we assess it to say what really happens to the lines there? Can our existing logical models truly carry us that far?)
In a case like the one implied in your quote above, I would simply say that we disagree about what is logical. I might say that a given argument doesn’t conform with logical principles, or even that someone is “not being logical”. I should explain my reasons for saying so, as may you, and we may or may not reach agreement. On the other hand, if we all get to use different “logics” at will, logic loses all meaning and function–and we lose any hope that it correctly reflects reality. I think a better term to use in such situations is “opinions”, which my spell-checker is not seeking to correct the plural of There’s a lot of other language we could use as well. We certainly can have different opinions about things, even logic. I hope that’s all logical