>>127159365a lot of 00s emo style does come from "real" emo. there's crossover with stuff like the used and thursday and taking back sunday which started out as emo bands in the hardcore scenes and then took off. then people associate that style with emo, and extend that to other bands with similar elements like hawthorne heights and mychem. eventually black hair and eyeliner=emo for the general public. you also have influences from nine inch nails, marilyn manson, and other 90s bands because thats what the people who would eventually start those 00s pop emo bands listened to.
at the same time you have lifetime->saves the day->new found glory->fall out boy pipeline which took a style of melodic pop punk with emotional lyrics to the mainstream, where it then intermingled and became associated with blink, good charlotte, sum 41, etc.
it all originates from hardcore. emo=emotional hardcore. mineral didn't really have any more to do with hardcore than fall out boy did in the early days. the difference is fall out boy had hits and changed their sound to broaden their appeal, so they win out in the public consciousness.