You compare the blockchain to the Internet, but I think this is wrong. The blockchain is an implementation detail (a very important one, but just implementation), while the Internet is just the idea. The implementation of Internet relies on TCP/IP. So, as I see it:
Bitcoin = Internet
Blockchain = TCP/IP
And, no, we shouldn't talk about Blockchain (which will create confusion with Blockchain.info, which is kind of brand now).
I agree with you. Saying 'put it on the blockchain' is even more confusing. 'Use bitcoin' is fine. I also think blockchain.info should have called itself something else as it is confusing, though I guess it was a smart move from a business perspective.