Basically what you're saying is that not all Europeans have been considered "white" at all times. Fine, just use the term non-white then. It's not like "people of color" is imune to any ambiguity. Are Iranians "people of color"? Why, or why not? It's just another silly term, not a panacea for race relations. I'm very much in favor of being objective, as that can never offend and rarely leaves room for misinterpretation or for "enlightenment flaunting". So if I mean people who are not white, I'll say non white.Except in a place like Australia there's plenty of white-passing people with non-European ancestry (I know people with Persian, Lebanese, Malaysian, South African Jewish, and Indigenous Australian ancestries who generally pass for "white"), and there's plenty of people with European ancestry who don't necessarily pass for white (a Greek Cypriot workmate of mine has experienced racist abuse in the street along the lines of "go back where you came from" that had to have been based on skin colour).
The term person of colour captures precisely what it is intended to capture - the experience and condition of being a visible ethnic minority in white hegemonic society.
Then there's the whole thing where for decades, Greeks, Italians etc were officially regarded as not white in immigration policy in Australia for decades, and then after that suffered a long period of racial abuse and discrimination.... so "white" and "European" haven't always been comfortable synonyms here.
I'd have thought this varied hugely by region and by which European language you're using. To my knowledge indio is a lot more accepted, common and self-applied in Brazil than in parts of the Spanish-speaking world and you can't necessarily assume it works the same as Indian in English. And of course I have no idea how people feel about indien in French Canada.
As for Indians, I admit my experience is limited to Brazil, a couple South American countries and the US. In these places, all Indians I've met referred to themselves as Indians (or the equivalent translation, like the Portuguese índio). I'm also 99.99% sure that the PC term "Native American" was invented by some white US-American college professor.