What are you arguing? Because the US is not the authoritative source for the names of Lake Ontario or the Gulf of Mexico.
Except it very much is. Within any locale the government dictates names. Now the US government is not the authoritative source for what something is called here where I am right now, which is why it still very much says Lake Ontario when I look at in on Google. Governments maintain databases of naming systems for all navigation. The USA government has multiple such, defer to states for some. In this case it's the GNIS managed by the USGS. They are the authoritative source for USA based landmark names. You may not like it (and yeah fuck authority and all that) but that's the way it is. The GNIS has changed the name so any map based in the USA not using the updated Lake America is objectively wrong.
The US government is not an authoritative source for the naming of bodies of water that are not within their control.
Of course it is for it's locale. In fact half of the lake belongs to the USA. You seem to be stuck with the concept that one place can have multiple names from different authoritative bodies. That's a thing that can happen, and in this case does.
The GNIS lists it as Lake America. Any map used in the USA that doesn't say Lake America is wrong.
The GNBC lists it as Lake Ontario. Any map used in Canada that doesn't say Lake Ontario is wrong.
It's not that complicated, especially for a system like Google Maps where naming has been geofenced by locale since basically it's inception.
Really the only difficult thing is what do you call something in a locale that doesn't define it, and Google's response to that has been pragmatic by listing all options (thought it seems they've yet to update Lake Ontario to follow the same naming convention as they applied to the Gulf which recognises there are two different *official and authoritative* names. )
Your last paragraph is quite apt; should the president of the United States be able to change the names of things that don't belong to the us, on a whim just because he feels like it? No.
I agree. But we don't live in a world of "shoulds". We live in a world of laws. The president should not have the authority. But he objectively does have it. Also he can't change the name outside of America, the ruling only applies to what America calls these places. That is it. Cross the border to Canada and Google maps displays Lake Ontario. Governments have always maintained the authority to call things how they see fit in their own locale. Hell sometimes that name sticks internationally and evolves through the eons. The Germans didn't call their country Germany. They called it Deutschland. The English didn't call it Germany either, Julius Caesar in Rome called them Germania in Latin, and the term was adopted by the Roman Empire.
The only reason to adopt someone else's naming convention is politeness. Do you still call it Turkey in English or do you use their preferred name Türkiye? I'm guessing you don't even know how to write an umlaut on your keyboard.