Saying "Bill pisses me off because he's white." is an entirely different conversation.

It's super fucking not, friend. If you define racism as something that doesn't apply to white people specifically, then you have a point, but "racism is when one is prejudiced against someone because of their race" is a much more usable definition than anything that involves socioeconomics, and I struggle to think of a better word to describe hating Bill for his race.

Prejudice + power describes institutional racism very well, but interpersonal racism doesn't necessarily involve power dynamics.