All of the above writeups are as correct as any can be about something as convoluted and broad as
emo nomenclature.
It is correct that music termed 'emo' or 'emocore' sprang up in Washington DC in the mid-1980s, when bands like Rites of Spring, Soul Side and Embrace began to play more mid-tempo music with lyrics centered around lost love, loneliness, confusion, and other topics usually reserved for bad teenage poetry. The "DC sound" came to be dual guitars, usually played in unison, with propulsive basslines, often carrying the melody or being a musical center point. The vocals were usually sung in a hoarse voice, not shouted like in the hardcore punk of the time, although usually at the song's climax there would usually be a breakdown into screaming of some sort. Usually when someone nowadays refers to this sound, it will be called emocore.
While emo started in DC, it spread all over the country shortly. Finding longstanding homes in the midwest and in California.
Hardcore Emo started in and around Silicon Valley, with bands like Mohinder, Heroin and, later,Portraits of Past and Swing Kids. These bands played short songs, usually sub-two minutes, with screamed and howled vocals. The guitars were distorted sometimes beyond recognition and were often just noise. Lyrics of ultimate pain and anger were prevalent, and releases generally include lyrics so you can figure out what the singer is doing all that hollering about. A special note should be made about Portraits of Past, who, after their initial release (a split 7" with Bleed), began incorporating longer, epic structures into their songs. This resulted in one LP of crushingly beautiful hardcore, complete with tense buildups and descending, breakdowns.
Emo incorporating pop song structures, and more singing/less screaming, started in the Midwest with bands like Split Lip. It was often much less angry, and more melancholy than really sad. This genre has become very popular, with bands like The Promise Ring and The Get Up Kids being well-known by everyone, even getting alternative rock radio airplay, and appearances on MTV. This has been much more notable recently (9/03/02) with the appearance of bands like Thursday and Dashboard Confessional on MTV constantly. I, as a snob and elitist, don't consider these bands to be emo. They're pop bands plain and simple. (I'm sure there are a million rants I could insert here about (welcomed and sought after) commercialism replacing any meaning and sincerity that music has, and they're all tired and you've heard them all before. I'll leave it at that.)
Classically (89-96, let's say, to be rough about it), "emo" was a style of music with heavy emphasis on loud/soft dynamics, and bursting tension at the song's climax. Vocals can range from whispering or speaking during the quiet parts to full on screaming, crying, or wailing during the loud parts. Bands like Moss Icon, Indian Summer, and Hoover are especially known for this. The music is more complex than your average three chord punk rock, but the musicians were very rarely virtuosos. The guitars were quite distorted, and octave chords were frequently used to give the music a rich tone. Many bands preferred a buildup of single-note melodies into a chorus or breakdown of wildly thrashing chords and drums, still anchored with a steady bassline.
Emo isn't supposed to be a subjective term for things, but it seems to have become one. Use your own judgement. What I may call emo, you may call hardcore and what you call emo I may call pop or indie rock.