Jimmy Eat World makes "emo" rock, one of those know-it-when-you-hear-it genres that seems as much an attitude as a type of music. "Emo" bands, which have done plenty of late to push amplified guitars back up the charts, tend to keep it simple instrument-wise, and they produce a grunge-inspired racket with a careful sense of verse-chorus songcraft. . . . Along with the geek-rockers of Weezer, Jimmy Eat World is probably the most popular emo band out there, with a couple of bona fide radio hits and a few critically acclaimed albums to its nonsensical name.
—David Segal. "Jimmy Eat World: Love And Angst Over Easy," The Washington Post, June 7, 2002

The opening Samiam semi-scored with a mostly grinding, intermittently soaring set, with singer Jason Beckout mixing rage and introspection and the band straddling the punk/metal line. "You can't exactly call it punk rock," said drummer Victor Indirizzo, after the set. "But it's in that vein. Some people call it 'melody-core' or 'emo-core,' for emotion."
—Jim Sullivan, "Bad Religion stays too faithful to itself," The Boston Globe, November 17, 1994

Here's the earliest use I could find for the shortened emo form:

This three piece does the loud, melodic emo (short for "emotional") punk thing kids seem to dig so much these days, accented by sudden speed shifts. —"1997 New Times Music Awards Showcase," Phoenix New Times, April 17, 1997