What you say about the US just being one of the American countries is correct, and it's often a source of confusion (intended or not)
You are right. But also the term Latino "creates" Latin America, which are the countries Latinos come from. And then also a subculture of immigrants in the US, which funnily is a bit excluding of Brazilians as not real Latinos (as there is also a expectation that everybody below the US speaks Spanish).
You should have stopped right there.
Latino is a US term to refer to people in the US by their origin.
Outside of the US people use Latinoamérica to refer to the central and South American former colonies of France, Portugal and Spain.
Iberoamérica is used too if you want to be more specific and exclude the French colonies
Oh yeah it's very funny when you start the "actually Quebec is part of Latin America, but the Guyanas aren't, except for French Guiana, which isn't actually a country, and about Louisiana...."
It's a bit of a complex issue since it's the US taking over a term or culture and normalising it as the only valid one.