:"Emo has a specific meaning, like banana, not an all-inclusive one, like fruit."  I could not possibly disagree more.

It's impossible to split the "historical" and "popular" notions of emo because they're the same thing.

Since my "grunge" example failed, here's another: "Hip hop".  In the 80s, "hip hop" described dance-influenced music that featured light-hearted lyrics, as performed by groups like Digital Undergound and DJ Jazzy Jeff and the Fresh Prince.  In the 90s, "hip hop" described Gangsta Rap, as performed by Snoop, Dr Dre, and 2Pac.  In the 00s, "hip hop" described an R&B-influenced; almalgamation of the two, as performed by Nelly and 50 Cent.

Your arguments that the original emo is the only emo is like claiming that Nelly and 50 Cent aren't "hip hop" because their music sounds little to nothing like the 80s version of "hip hop".  Instead, "hip hop" of 2005 should be remembered for some NYC underground band whose sound is reminiscent of 80s hip hop.

Here's the key: ask someone in twenty years what "emo" was like in 2005, what bands would you honestly expect them to throw out?  Indie bands that few people have heard of, or popular bands like Coheed and Cambria?

Splitting the scenes into "waves" is a perfectly acceptable way to reflect the popularity of "emo" over the years.  If you were to ask a random person what emo was in the 80s, they'd probably say Rites of Spring.  Ask a random person what emo was in the 90s, they'd probably say Sunny Day Real Estate or Jimmy Eat World.  (Don't disagree with this point simply because you would think of someone else first.)

Journalists and record labels didn't decide that the second wave of emo was emo.  Fans of the music did.  Regardless of how you feel about it, that's what happened.  It's historical and 100% true.

No matter who deemed modern emo to be emo, fans have accepted it.  End of story.

Encyclopedic articles (and Wiki articles, according to THE OFFICIAL GUIDELINES) are supposed to reflect the historical record shared by the most people.  Only a modest handful of people have heard of bands like Hot Cross and Trophy Scars, no matter how much you wish otherwise.

It's perfectly acceptable to include references to bands that still practice original emo.  That's a worthy part of the historical record.  However, your insistence that modern emo is illegitimate and should not be covered in the same article is a minority POV, which the Wiki rules are explicitly against.

Seriously, don't get me wrong: I'm entirely frustrated that "emo" means something else now than it did ten years ago.  Every time I mention to people that I was in an emo band in the late 90s, I basically have to defend myself and go into some long-winded explanation that's completely meaningless to them.  BUT THAT'S LIFE.  That's how things are now, and how the word "emo" has shifted in the last twenty years.  They know "emo" to be something completely different from what it was, and I have to live with that.

And so does everyone else working on this article.  ChrisB