Fugazi actually is Emo. As are Rites Of Spring.
The problem, however, is that the term 'Emo' got too closely associated with bands like Sunny Day Real Estate during the mid-90s, after the original wave of Emotional Hardcore (abbrev. Emocore, Emo) died out. What was left behind was actually Post-Emo Indie Rock, but nobody wanted to say all that, and when so-called 'Emo' became popular, it was SDRE, Weezer, and the scores of bands that came out afterwards that imitated them or infused it with Pop-Punk.
I think the At The Drive-In/Mars Volta (what, no Sparta?

) thing is that now people are just calling any grand sounding, punkish rock music with wailing vocals Emo, regardless of the lyrical content. I prefer to lump them into the Prog Rock category, myself.