The problem here is that the "primary, common, local" name used in a country sometimes conflicts with what specific government institutions in those countries may "officially" use.
In the situation with the Gulf, it's unambiguously true that the "primary, common, local" name used in the US by English-speaking Americans is "Gulf of Mexico", but they decided to adhere to an artificial name change declared by a political figure, whose authority to make that "official" only extends to a single branch of a single level of government.
In the situation with the Gulf, it's unambiguously true that the "primary, common, local" name used in the US by English-speaking Americans is "Gulf of Mexico", but they decided to adhere to an artificial name change declared by a political figure, whose authority to make that "official" only extends to a single branch of a single level of government.