In particular, the route to recognizing, and explaining, something like moving the goalposts from first principles can be kind of long and convoluted. Having vocabulary for it helps a lot.

While I agree with the notion that in many fields it helps a lot to agree upon a common vocabulary for speed/shorthand, accuracy and clarity in communication, in this case I think that learning the vocabulary is learning the wrong thing.

Shorthands also have a couple of obvious pitfalls. One of them being in this case that neither party might recognize particular leakiness in a generally agreed-upon abstraction.

What you (IMHO) should be learning--and the vocabulary sort of comes implied with this, but is not necessary--is how to recognize the shape of such a concept or structure (in this case, fallacies) and how to "disarm" them, deal with them, in the context of an argument. Pointing out the fallacy by name should only be done maybe after disarming the fallacy in context of the argument, as further clarification.

If doing that is too long and convoluted, it really goes to show that learning a common vocabulary is training for the wrong thing. Instead, train to make your ways of dealing with fallacies shorter and to the point.

Say you know and studied all the moves in karate; During a match, your opponent comes at you with a certain attack, and you know the perfect defence / counter attack to it because you studied it all. You call out the name of your opponent's attack and shout the name of your counter attack. Having agreed upon a common vocabulary, your opponent knows they are defeated by your counter. A shouting match ensues.

I'm not saying that having a common vocabulary is not useful, it's very useful. Imagine two karate trainers sharing tips on what techniques best to learn students of a certain level. Imagine how they are even able to talk about "students of a certain level". But that's all meta game. In a real fight the moves have no names, and in a real argument neither do the fallacies.