At least in Bosnia, there are no variations of the börek. There is only one börek - with meat. Other "variations" with cheese, potato etc.. they have also a different name. Basically if you order a börek, you will get it with meat.
Yes, this is a common "fight" between people in Bosnia and neighboring countries:
In Bosnia, "börek" (or "burek", as is its local spelling) is only filled with meat, while the rest of the dishes fall under the broader category of "pitas" (or pies): sirnica (filled with cheese), krompiruša (filled with potatoes), zeljanica (filled with cabbage), jabukovača (filled with apples)...
In Serbia and Croatia, they're all "burek" with different stuffings, so burek with cheese, burek with potatoes, etc.
It's a weird hill to die on, but you will receive some weird looks if you request just "burek" in Serbia/Croatia/Montenegro, or "burek with cheese" within Bosnia.
Zeljanica is, despite its name, filled with spinach rather than cabbage. Then there's also an abomination called "pizza-burek", it's probably as bad as it sounds...
There's a few places in the Chicagoland area that serve Börek (I see it spelled Burek here also). The one closest to me offers fillings of ground beef, spinach and feta, mushrooms, or potato. They are all very good.