“In English, the unqualified term "America" often refers to the United States of America as a synecdoche, with "American" typically referring to people and things from that country. […]”
Regarding the gulf naming, “Gulf of North America” would seem somewhat appropriate, if a mouthful.
I disagree about that specific point. It’s one of the meanings of “America”, and is used that way throughout the world. From https://www.merriam-webster.com/dictionary/America:
1 either continent (North America or South America) of the western hemisphere
2 or the Americas the lands of the western hemisphere including North, Central, and South America and the West Indies
3 UNITED STATES OF AMERICA
Similarly in Wiktionary: https://en.wiktionary.org/wiki/America#Proper_noun
“In English, the unqualified term "America" often refers to the United States of America as a synecdoche, with "American" typically referring to people and things from that country. […]”
Regarding the gulf naming, “Gulf of North America” would seem somewhat appropriate, if a mouthful.