Regarding the specific example of the woman who said "I eat salad", I would say that white privilege has something to do with this, but it wasn't the main factor. I think any white privilege component would be largely unconscious, e.g. she might have felt more emboldened to speak in an insulting way since they were black and probably, in her eyes, more uneducated and inclined to be more rude anyway.
I would say that her rude behavior has more to do with her being an assertive and obnoxious person by nature. She would probably be rude as well if the two people were white, but I believe that she would not have been *as* rude, and this would be due in significant part to white privilege.