I'm super sympathetic to all of this.

In the special case of justification, part of the reason I found Williamson's intervention in debates between internalists and externalists about justification so interesting was that it purported to rule out this kind of response, by showing that the internalist concept of justification is empty (because nothing is operationalizable in the way they demand justification must be). Whether it succeeds is another question, but if he does, the internalist can't say "we're talking about one concept useful for one set of purposes, and they're talking about a slightly different one, useful for a different set of purposes." To continue the mathematical analogy, it would be more like different conceptions of set, one of which let's you construct Russel's paradox; that one is just ruled out!