Sadly, in the US, we have a terrible system where schools are locally funded, and poor people get poor schools. Under this system, socioeconomic class self-perpetuates.
I think you've just described racism.
My point is that the resulting behaviors and discrimination can still be racist, even if they don't have their ultimate roots in racism itself.
From your link:
> also : behavior or attitudes that reflect and foster this belief : racial discrimination or prejudice
> : the systemic oppression of a racial group to the social, economic, and political advantage of another
Edit: Anyway, I do recognize that the distinction you're pointing out does have implications for the best way to address the problems.
I hope our discrimination problems fall more along class lines, because that's much easier to solve by implementing economic policies to reduce economic inequalities. Government programs won't change people's races (well, non-terrifying government programs, at least), but policies can change people's economic class.
It's also entirely possible that different areas of the country have different mixes of classism and racism. My wife is of a different race, and our son is mixed race, and my wife and I both went to top-tier schools, so my social group is almost certainly less racist and more classist than the US median. It's very difficult and dangerous to extrapolate my lived truth to the whole nation.
Every class tranche uses fashion to differentiate themselves from the tranche just below, while imitating the tranches above. The upper class used to wear fancy clothes to differentiate themselves from the middle class; the middle class started wearing the clothes of the upper class to try and join them; boom, ghetto chic becomes popular among ivy-leaguers. (They don't have to worry about being mistaken for actual lower-class people; the cultural differences are big enough that that isn't a concern).
This helps maintain in-group selection with all the wonderful effects it has.