Your claim that the American understanding of “rightism” isn’t the true understanding just points out your fundamental lack of realization that we’re talking about two different political axes here. The term “right” isn’t used in the US in the way that you seem to believe it is. Trying to conflate the European “right” with the American “right” is nonsensical.
I made no such claim. American understanding is tilted toward republicanism and is often confused with libertarianism, but the "right" means pretty much the same thing everywhere: individual autonomy under law, strong national institutions, government limited to points of national interest, pursuing them vigorously.