Though it has been defined by many sources - Wikipedia, Andy Greenwald, Andy Radin and such - as a subgenre of punk music in the late 80's, we have to consider that in this modern era, Emo is now defined as anything overly emotional.
Associated with this are self-mutilation, skintight clothing, piercings, excessive crying, and I could go on for days. There is no exact, bullseye definition for Emo since it is somehow stuck between being a genre of music, which is what it used to be and; a short-cut term for the word "emotional".
But if we talk about now, Emo means emotional. Cutting your wrists, jumping off your window, listening and crying to "emotional" music, drugs, sad relationships, guns, razors, sharp objects, blood, hating, rationality, loving, sex, homosexuality, breaking up, and so much more.
It's kind of nonsense to some, but what can we do? It's there and if you don't like it, you'll just have to wait for it to die out.
It is emo to cut yourself.