The President of the United Stated doesn’t actually have the authority to tell anyone besides the executive branch what to name geographic places. So the order only forces official US government maps created by USGS and the like to be updated. Other maps can choose to update if they want to.
Open Street Map tends to reserve the “name” tag for what things are commonly called by the local community. It appears that for the Lake Ontario and The Gulf of Mexico they’ve added an “official_name:US” tag to capture the names in the executive orders.
They tag according to global consensus, so they use Gulf of Mexico in the 'name' field. There is an attribute for 'official_name:en-US' that returns Gulf of America. They do the same with other disputed names/places.
Well, when you visit it https://www.openstreetmap.org/#map=7/23.765/-87.100 there's no naming of that area at all. At least when looked at from a US IP. And when you search "Gulf of America" it returns "Gulf of Mexico". So take from that what you will.
Also https://www.openstreetmap.org/#map=8/43.622/-78.852 shows "Lake Ontario". But it does carry the tag "official_name:en-US": "Lake America", so search will find Lake America and a mapping app could decide to show the official name
Though generally official names are the names nobody uses. For example the country labeled "France" also has the attribute "official_name:en": "French Republic"
Open Street Map tends to reserve the “name” tag for what things are commonly called by the local community. It appears that for the Lake Ontario and The Gulf of Mexico they’ve added an “official_name:US” tag to capture the names in the executive orders.
https://wiki.openstreetmap.org/wiki/Key:official_name:en-US
Though generally official names are the names nobody uses. For example the country labeled "France" also has the attribute "official_name:en": "French Republic"
If they ask what the rules of the game are, they've just lost again.
They didn't invent being deliberately obtuse.
Now, I'd love for each geographical region to take their very worst area, and rename it after a certain politician for the lulz.