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).