So "both" means a) the endonym and b) the exonym of the country in which the user is currently located? Makes sense, I guess.
Edit: However, as far as I can tell, there IS no endonym in this case, so it should be exonyms all round?
Edit: However, as far as I can tell, there IS no endonym in this case, so it should be exonyms all round?
So yes, there are only exonyms.
I'm not aware of any other places where there is the case in English but looking at places in India it seems the exonym just appears on a different line above the endonym.
So I would expect to see
Gulf of Mexico
Gulf of America
over the body of water when this is updated.