The emerging consensus is that there is now an alternative English name, and that this is likely to end up as an official name for en_US (which can either mean American English, or the English name according to the United States) as well. It seems unlikely that the generic endonym (Golfo de México) will change any time soon, and unless this English alternative is adopted more broadly, it is likely that for the foreseeable future the English name too will remain 'Gulf of Mexico'.
What won't happen, is that visitors from the US will see that new name, at least in most OSM apps and websites. OpenStreetMap is not limited to one renderer or app, and anyone who builds one can choose how to handle localisation. But to support this name based on the tags proposed now such an app would have to prefer official names over common names, or explicitly choose this local alternative over the common name.
This is what the relevant name tags could end up looking like soon (other languages omitted for brevity):
name=Golfo de México
name:en=Gulf of Mexico
name:es=Golfo de México
official_name:en_US=Gulf of America
alt_name:en_US=Gulf of America
Depending on how this all pans out, at some point name:en (the global English name) could shift to the Trumpian name, but that is a big if of course. OpenStreetMap simply documents the status quo, and that demonstrably isn't it right now (and might never become it). If at some point the new name becomes the common name in American English, it might end in name:en_US as well. Feeling this out would be mostly up to the American OSM community (e.g., do most media call it that now? How about US scientists and relevant institutes? What about use on the internet by American folk? Etc.)
A really simple map might only render whatever is noted in name. A map which takes the user's locale into account, or which explicitly chooses a base language or locale, would use a cascading fallback, which for en_US looks like:
* Try name:en_US,
* If missing, use name:en
* If missing, use name
So for now that would not yield 'Gulf of America', because usually neither alternative nor official names are considered for rendering.
In any case, the sea can be found using search under all those names.
This is an excellent, detailed comment — it's so good I want to ensure it's exactly as you intended. In your "This is what the tags could end up looking like soon" example, should at least one of those (alt_name:en_US?) be "Gulf of America"?
Most likely add an American English name, leaving the English name as Gulf of Mexico.
But I didn't read all 108 comments in the discussion.