It's a very lawyerly distinction you're making here that's willfully blind to what's really going on. The common-sense reading "primary, common, local name" is that it should be called the Gulf of Mexico, as that is what it is called in the USA.
If the nutso president of Turkmenistan decided one day that the Caspian Sea's real name was the Turkmen Sea, there's absolutely no way that Google Maps would rename it on their maps. The reason they renamed it in this case is not because "well, the government said so, so we have no choice!". It's (obviously!) done to not piss of Trump, and pissing off Trump would be bad for their business. They can hide behind lawyerly arguing all they want, but that is the real reason.
If the nutso president of Turkmenistan decided one day that the Caspian Sea's real name was the Turkmen Sea, there's absolutely no way that Google Maps would rename it on their maps. The reason they renamed it in this case is not because "well, the government said so, so we have no choice!". It's (obviously!) done to not piss of Trump, and pissing off Trump would be bad for their business. They can hide behind lawyerly arguing all they want, but that is the real reason.