In mainstream American lingo “the Rebbe” is talking about the Lubavitcher Rebbe. Secular Americans don’t even realize “the Rebbe” could mean someone else. I’m not endorsing the trend, just recognizing it. Never ever, would an American be referring to the Gerrer Rebbe by saying “The Rebbe.” I’m not sure where people got that idea.
I’ve never heard anyone leave their community and refer to their communal Rabbi as “The Rav;” they always say “our Rav,” or “our Rabbi.” Within the shul, yes, but not in all circumstances.
I remember going to the OU homepage a few years back and seeing it just plastered with stuff about “The Rav,” referring to JB Soloveitchik. I assume it was to compete with the very successful Chabad website which is decorated in a similar fashion. Why did this thread get bumped so many years later?