In short: both are listed, but the primary name remains.
the OpenStreetMap Foundation’s longstanding policy regarding geopolitical disputes is to prioritize the on the ground rule by default but also record information about notable alternative claims, so that data consumers such as renderers and geocoders can choose what to present to end users.
OSM is mostly maintained by EU/German grants right?
I don’t know about funding, but OSM is user edited and has no doubt reverted the name many times now:
For convenience:
https://community.openstreetmap.org/t/gulf-of-america-gulf-of-mexico/124571
https://wiki.openstreetmap.org/wiki/Gulf_of_Mexico
In short: both are listed, but the primary name remains.
Idk, it’s just what I use for everything except satellite imagery.