I'm not so sure on the Turkmen Sea example. Google is (trying to) sidestep this issue as much as possible by making the name on the map depend on your physical location, language and localization settings, or both. They could easily make the Caspian Sea say Turkmen Sea only in Turkmenistan, and probably would if they wanted to keep doing business in Turkmenistan. This is already how it works for Lake Ontario: if I access from the US, it says "Lake America", but if I VPN to somewhere in Europe, it says "Lake Ontario (Lake America)".

In terms of lawyerly distinctions, I would actually argue that there is still no legal basis to the rename. The president has directed the federal executive branch to refer to Lake Ontario as Lake America, but...that's it. It literally doesn't and can't apply to Google. No law has been passed, and even if there had been it's pretty clear that it's a 1st Amendment issue to compel private entities to refer to things in particular ways -- it would be a slam dunk even on this Supreme Court. Their decision to do it is a voluntary act meant to curry favor with this administration, or at least to avoid pissing it off. This isn't like the Indian laws that make it illegal to publish a map that shows Kashmir as part of Pakistan, it's just a memo to the federal bureaucracy. Yes, this doesn't really matter because there are punishments that are within the power of the president (like cancelling contracts, or cooking up bogus DoJ investigations, or failing to resist antitrust enforcement, etc.) that they want to avoid. This just goes to illustrate that the rule of law in the United States is mostly a fantasy.