There’s a slight difference in that “Ostsee” is the common name. If the German chancellor decides to call it “Deutsche See” tomorrow, the name would continue to be “Ostsee”, because that’s how language usually works.

“Gulf of America” is just a dictator’s wish of a common name. The people of OpenStreetMap decided to use the tag “official_name:en_US” for that reason, while keeping “Gulf of Mexico” for the commonly used “name:en-US”.