*Frog Dissection Ahead*
Gezhe is a joke. It's a term whose definition changes depending on the context and who's using it. It generally means OG Lubavitcher from specific Chassidic towns. It's not something 99.9% of Lubavitchers take seriously. The most common current usage, outside of jokes, is when relative newcomers think they aren't getting a seat at the table because the "gezhe" are Deep State and control the game. You're getting a peak at 50 years of gezhe jokes.
There are certainly deep state who in Lubavitch who are not Gezhe.
Here's a map. I'm sure many towns might be recognized. But it's only the few (Nevel, Klimovitch, Beshinkovitch, Dokshitz and maybe 1 or 2 more) that qualify for the true gezhe hallmark.

ETA: just for fun I've expanded the map to include the (Lithuanian) town where my maternal grandmother comes from, which actually had a Chassidshe community, but my grandmother's family wasn't part of it.