[Beretta_Vexee]

They probably mean endonymy and exonymy.

https://en.wikipedia.org/wiki/Endonym_and_exonym

Different groups can have different names for the same thing. For example, country names change depending on the language.

[oneeyedpigeon]

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?

[Beretta_Vexee]

It's a bit complicated to talk about an endonym for the Gulf of Mexico. There's no endogenous group in the Gulf of Mexico, because nobody lives there.

So yes, there are only exonyms.

[mathieuh]

The endonym and exonym in English were the same previously (Gulf of Mexico). The endonym is now is changing to Gulf of America while the exonym is remaining Gulf of Mexico.

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.