The Rav and The Rov are both spelled and pronounced differently I don’t see the confusion. The first time I heard someone mention The Rav I didn’t know who he meant but the pronunciation made it obvious to me that he did NOT mean The Brisker Rov.
When Rav Ahron Kotler was alive “the Rosh Yeshiva” usually meant him unless you were in the actual Yeshiva or talking to another Talmid of someone else. Afterward he was nifar if that distinction was used it was a reference to Rav Moshe Feinstein. I can’t think of any Rosh Yeshiva alive today who is referred to as “The Rosh Yeshiva” by non Talmidim.