No, it doesn't. Quoting the document: "If different countries dispute the proper name for a body of water, our policy is to display both names, with each label placed closer to the country or countries that use it" (emphasis in original).

This sounds to me like for all users it should display both names. Perhaps to US users it displays "Gulf of America (Gulf of Mexico)" and everywhere else it displays "Gulf of Mexico (Gulf of America)", although the literal text suggests it should say "Gulf of Mexico" in an area of the map closest to Mexico and "Gulf of America" in an area of the map closest to the US.

Incorrect. They are explicit that they use the 'common' name, specifically to avoid arbitrary changes by (despotic) rulers.

"everyone else sees both labels"

Just because the US has decided to smear faeces on their walls doesn't mean we all need to admire it.

It sounds fucking stupid for this irrelevant name to be shown to non-American countries.