While this specific rename looks dumb, the map providers deal with this kind of things all the time. Big georgraphical features like rivers, lakes, seas, mountains and also borders are different from different perspectives (mostly country-based).
From outside of the US, I see "Lake Ontario (Lake America)" in Google Maps, the same way as with the Gulf of Mexico. I see "Persian Gulf (Arabian Gulf)" and "Sea of Japan (East Sea)".
Some names are given from one perspective (like rivers and mountains), like Rio Grande river or Mt Everest.
From outside of the US, I see "Lake Ontario (Lake America)" in Google Maps, the same way as with the Gulf of Mexico. I see "Persian Gulf (Arabian Gulf)" and "Sea of Japan (East Sea)".
Some names are given from one perspective (like rivers and mountains), like Rio Grande river or Mt Everest.