Residents of Upper Canada participated in the War of 1812. But more to the point, the notion of a clear distinction between persons of "British" and "Canadian" identity respectively is an anachronism, in this context. English Canadians (and in particular, the ruling class) overwhelmingly were British, and regarded themselves as such, at this point in history. The notion of a Canadian nationality which expressed itself independent of or even complementary to that of British territorial identity does not weigh upon the situation.

So on one level, it must be granted that speaking of "Canada" having burned the White House is a bit silly, to the extent that turn-of-the-19th-century Upper Canada possesses only a distant and complex ancestral relationship to the modern Canadian polity. But on another level, there is perfectly good sense in acknowledging that a "Canada" (i.e., Upper Canada) was relevant to this contemporaneous context, and attributing the war of 1812 to that polity, as a territory of the British Empire, is not at all unreasonable, given that it both served as battleground, and its residents, along with its British countrymen from elsewhere in the Empire, participated in the fight.

Our notions of the continuity of nations (i.e., when a nation is "born", or when it yields to a new identity) are often rather arbitrary, and a function of traditional narratives.

But in this case, choosing a solution which elects to erase Upper Canada and its people from existence, rather than acknowledging them for what they were, and the role they played, strikes me as daft.