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 think the OP is seeking an acceptable term to express the myriad cryptos now and coming down the pike that are based on blockchain technology, for referring to them as Bitcoin won't do them justice. At this penning, the best I'm able to ascertain is bChain, wondering if it, or something like it, fits the bill.