The Internet is the network of networks, connected and running the TCP/IP protocol. That's how I usually use the term. looking at my dog-eared copy of The Whole Internet User's Guide and Catalog by Ed Krol (O'Reilly, 1992), I see his definition may be a bit broader. He says:
'Five years ago  the answer would have been easy: "All the networks, using the IP protocol, that cooperate to form a seamless network for their collective users." ... More recently, some non-IP-based networks saw that the Internet was good. ... [T]hey developed methods of connecting these "strange" networks (e.g. Bitnet, DECnets, etc.) to the Internet. ... Are they part of the Internet? Maybe yes and maybe no.' (p.13)
So the definition could be braoder and include networks running TCP/IP connected to each other and the other networks connected to them, seamlessly from the user perspective.
An internet, is a network running the the TCP/IP protocol. It may or may not be connected to the Internet (and a part of).