I recently added the following shortest definitions for Internet, Intranet and Extranet in their Wikipedia articles, though they are not new concepts even out-of-date sometimes:
"The Internet can be briefly understood as a network of networks."
"Briefly, an intranet can be understood as a private version of the Internet."
"Briefly, an extranet can be understood as a private internet over the Internet."
Additionally, an internet (lowercase i) refers to any interconnected networks, including those not based on TCP/IP protocol suite. Therefore, an extranet can be referred to non-IP networks (such as IPX/SPX networks) interconnected over the Internet via VPN.
A short definition does not mean its connotation is "short" too, right?
For the above four definitions, the "Internet" one was what I learnt; the "Intranet" and "internet" ones are my personal understandings though some people say so as well; the "Extranet" one is newly defined by me, hence there is no people saying the same today, at least on the Internet covered by Google. :-)
Giving a brief definition actually needs a better understanding, and commonly makes it more easy to be understood by other people.
I like using brief definitions.