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).