The correct name depends on your context. I know it’s unpopular but it’s true.
And the correct name in all contexts that matter is "Lake Ontario".
It's almost as if people don't know that "Ontario" is not a Canadian name, it's a native American one, and the province is named after the lake, not the other way around.
It was called "Lake Ontario" before Canada or the US even existed.
I say that there are two lakes. It remains for future generations to decide how the two lakes are apportioned. Perhaps each of them are coterminous, with the same boundaries, size and volume, but one is named "Lake America" and one is not. One is a lake in US waters, and the other is in some foreign country.
Perhaps of the two lakes, they are split in twain, and there is an international border running through the middle of these two distinct bodies of water. Perhaps vessels will littorally obtain right of passage in international waters, and undergo customs inspections before they can cross over the somewhat amorphous border between Lake Ontario and Lake America, and vice versa.
2) You really had to choose that country? https://en.wikipedia.org/wiki/East_Germany
If Trump tells Google, and spends taxpayer money, to "officially rename" Ottawa itself "Poopietown", then should they do it?
Maybe we need a constitutional amendment to remove the president's power over place names, given that the founding fathers only considered a malevolent president, not one with the brain of a 5 year old.
I remember years ago driving across the border at Niagara Falls, and just being waved though, no passport required, when I said I was just on a day trip.