I honestly feel like 80's post hardcore (Fugazie, Rites of Spring, Jawbox, Husker Du, etc.) should be considered separate from modern day post hardcore (At the Drive-In, Silverstein, Chiodos). Just like how late 70's/early 80's post punk (Wire, Joy Division, The Cure, Public Image Ltd. etc.) are considered separate from post punk revival (Interpol, Franz Ferdinand, Arctic Monkeys, etc.)
Maybe Post Hardcore Revival could be a better term?
not necessarily emo, bands like At The Drive-In, Fugazi, Big Black etc. combined hardcore punk with heavier bass and faster tempos. But I will say that most post-hardcore from the 21st century onward is emo.