Early 'emo' bands are what we call indie bands now. Weezer/Modest Mouse/Cursive. The word has simply been recycled so many times that it's easy to say that Weezer has nothing to do with Emo because they sound nothing like From First to Last, or they have nothing to do with emo because they sound nothing like Simple Plan, or " " nothing like Motion City Soundtrack. To be truly correct about any of this, you have to look at the changes in the music and the term that connect all of them.
But that's my point, "emo" is a crap title for a genre and saying anyone other than the original dc guys are "responsible" for it is just more confusing. It's a useless tag as a genre and should not be used for precisely the reason you stated. You are right that Weezer influenced a lot of pop, rock, and "indie" groups and that Weezer had obvious influences from the emotive hardcore stuff, but to inject the term "emo" is not helpful at all at this point. They weren't called "emo" at the time, and they're not what we call "emo" now.
It's not a matter of "I like weezer and I don't want them associated with emo, cuz emo is lame" it's just not a useful comparison.