No, that was never the debate. The question merely centered around who could claim the language that implied that their particular CAUCUS was the default Church.
Merely?
I'm guessing:
There are two ways the question will be addressed. Some will take what you are calling the Augustinian Visible/invisible view. To them, I would think, NO organization could call itself Catholic (or Orthodox or any thing other than about comparative accidental matters like Anglican, Lutheran, Wesleyan .....
Others would take a view sort of in our ball-park: that while the furthest boundaries of the Church are not known, and wheat and tares grow in one field, still some visible shoppe can be thought of as the Full or blue-ribbon or sho' 'nuff real deal Church.
So there will be two arguments:
(1)Are the "invisible" people right; and
(2)if they're not which of the many contenders for "read deal" is right?
This question will not be settled any time soon. So in the meantime there is the question of what to call the Caucuses. Nobody will agree with all the designations of all the caucuses, and since we make the most outrageous claims we will take the most hits.
I think to call the question "mere" is optimistic.