I think this is a great opportunity to tidy up the names.
"Flat Earth" is quite vague - it could be flat and square, for example.
Similarly, "Round Earth" can also be Flat Earth, if the Earth is a disc.
I appreciate the difficulty describing a flat map of the earth which agrees with cosmological, navigational and astronomical evidence, but in the absence of such a map the best the "Ballies" have to work with is the "Flatties'" own logo.
"Globers" and "Discers"? C'mon guys, you can think of something funnier than that