Emo is ONLY an insult given by Punks directed towards the avid fans of Emotive Hardcore Punk & the genre itself. Emotive started in the mid/early 80's in the D.C. Hardcore scene. Emotive is essentially watered down Hardcore Punk. This type of music wasn't accepted in the scene, thus the term Emo was born.
2 of the most noted bands to take this form of Punk in a new direction, was Rites Of Spring & Embrace. Guy Picciotto, the lead singer of Rites Of Spring; most famous for his work in Fugazi, paved the way for what we now know as Emotive Hardcore Punk. Ian MacKaye, most famous for his Hardcore Punk band Minor Threat & the creation of Straight Edge; was also was a defining band.
Emotive in the '80's resembled it's parent genre, with more emotionally charged lyrics & a new tone. But in the '90's Emotive took a totally different feel.
Emotive in the '90's took a more Indie feel to their music. More commonly known as Post-Emo, it was bands like Cap'n Jazz, Jets To Brazil, Joan Of Arc, Sunny Day Real Estate, etc etc... Cap'n Jazz helped transform emo from a deeply underground Punk sub-genre, into a more widely accepted subset of Indie rock.
Emotive in the '90's also gave birth to what we now know as Screamo (Skramz). Screamo (Skramz) is somewhat of a misnomer. Though the genre often had screaming vocals, it wasn't the focus of the genre. Skramz is a more chaotic way of playing it's parent genre.
Rites Of Spring is Emo.
I Hate Myself is Emo.
Dashboard Confessional is NOT Emo.
My Chemical Romance is NOT Emo.
After School Knife fight is Skramz.
Toru Okada is Skramz.
Avenged Sevenfold is NOT Skramz.
Bullet For My Valentine is NOT Skramz.