"Slut" and "whore" describe behaviour, so I'd say that's rather different. I don't know about "tart" – I can't recall ever using or encountering it, except for one film (where a guy calls guy bloke a "tart").
Look, I don't want to tell you how to perceive words because that's always a personal thing, but in huge parts of the world it's just a strong but generic insult. And in the end context and intent always matters, not words. "You dirty person of colour" is of course profoundly racists in spite of not using any racial epitaphs. It's not the words themselves that are problematic, it's the intent with which they're used. It seems pretty clear the intent of my comment was not sexist.
