"The Internet" is the big public internet.
So you connect your internet to the Internet.
This has nothing to do with what the legal meanings will be, since they'll be drafted by people who either don't understand or understand perfectly and are maneuvering for advantage.
The thing that makes something the Internet, as opposed to a network using some random set of IP addresses, is that you have your IP addresses assigned to you by the Internet authority. You don't have to be connected to the big global network to be part of the Internet.
