I think that "emo" is more of a personality/style than a music genre. Because if you think about it, you get "emo music" from a band. If the lead singer (or whoever writes the songs) is emo, they write about emo things. Which makes the music not really emo, but it's the person themselves. Somebody who is depressed a lot and has a lot of emotions can be considered emo. It doesn't mean that they have to cut themselves, but some do.
That's just my opinion!