It's up to the renderer what to display. OSM tags data according to global consensus, they use Gulf of Mexico for 'name'. There is an attribute for 'official_name:en-US' that returns Gulf of America. Same approach for other disputed names/places.

https://wiki.openstreetmap.org/wiki/Key:official_name:en-US

"Few if any renderers consume official_name=* or any of its subkeys; in practice, the purpose of this key is for geocoders to find the feature when a user searches for the official name."