What's wrong with just saying "We are a media company" if you are a media company. The definition of a media company is fairly well understood, right? Why insist that you occupy some slightly semantically different space?
Context man! He was speaking at a media conference filled with traditional media co's, such as The New Yorker, Viacom, and ESPN. It was even run by WSJ. To that audience, his statement highlights the nuance of traffic referral vs. accumulating page views, which is why Twitter is potentially poised for greater monetization than a "traditional" media company.
It feels unfair to pull that quote out of context. The point you are making is very different than the point that Costolo was making when he said it.
Do you say you are a media company? Because - by your definition - you are! There are plenty of media companies that don't sell advertising (eg, book publishers).
Twitter isn't a media company because they say they aren't! They see their goal as microblogging/self expression/communication (or whatever it is), and selling ads is just a means to that end. Google is that same - they see themselves as a engineering company organizing all the worlds information, and selling ads is just how they pay for it.
Compare that to something like News Ltd or NYT. They seem themselves as a media companies, and so when they run into technology problems they don't adapt well.
