In recent posts here and here I have proposed that Facebook, Google, and Groupon are advertising companies because their clients pay them for advertising. I argued that they use the Internet and technology to do what they do, but that calling a company an Internet company is soon to seem very old fashioned. Would you call General Electric an electricity company? Would you call the New York Times a printing press company or an Internet company, or a news company? (I think news because they still do have subscribers that buy the paper, but advertising is right in there too.)
Where does Twitter fit in? Right now they don't seem to be getting paid for anything. It is likely that their monetization scheme will be advertising. Whatever they pick, I am sticking with my assertion that companies serve the people that pay them. Before long Twitter will be an advertising company too.
Here is a post from some time about about Third Party Payers. True, there are many industries with third party payment systems and most of them are broken. Companies would be smart to recognize that they serve their true masters -- those that pay the bills.