Argentina, ...and Uruguay I don't even need to explain. They're obviously western in almost every sense of the word. If you count them out because they've had dictatorships or because they used to have a few mestizos, you'd have to count out Italy, Spain and Portugal from the western world.
Except, every sense that matters ) They are not part of any Western alliance or organization, they are not closely linked to the West by trade, Argentina is not in any sense a liberal democracy (and hasn't really ever been one). Their only claim to being Western is the skin color of (most) of their inhabitants. But on that count Russia should be considered part of the West - and I haven't seen anyone here who does believe that.
The superficial ancestral Westernness of the Argentinians, etc., is just that - a long-decayed heritage. The polity these Italians/Germans/Slavs created in the New World has developed in very exotic ways, under very little Western influence.
It depends how you define the West really. Under your definition Latin America certainly isn't Western. In my view Western means anything relating to European culture which is defined as the Germanic/Latin/Slavic cultures that have developed thousands of years ago but also takes into account politics, somewhat, and economics, somewhat. Under my definition most of Russia is Western, large portions of the Balkans are Western, Greece is Western if only because it gave us most of our political and philosophical traditions. Israel is arguably Western because its populace has been so influenced by Europe in the past and have carried on democratic traditions which are so prevalent in Western tradition.
So its all relative really. I think it makes sense why a Spaniard of the past wouldn't view his nation as Western.
Why do you view the West as such of a political term? To me it has always been more of a cultural and economic term? After all the West has had its bouts of authoritarianism and has had big lapses in its liberal democratic tradition. I think your definition is a little too narrow and focused on diplomacy and politics. Mine might be too broad.