That's ridiculous! How do you rename half of a lake!

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.

That's not really how local naming works. I call Germany Germany but the Germans call it Deutschland. There's not two Germany's, there's just one and I have a different name for it. It really doesn't matter if a geographic object is completely inside, partially inside, or completely outside a country's borders, they have an official or de facto name for it that could differ from what you call it.