EU justice commissioner Viviane Reding is expected to table a draft bill on Wednesday giving individuals the right to ask for data about them to be deleted from the Internet unless there are "legitimate" grounds not to. The new law as part of an overhaul of the Data Protection Directive.
I think the latter is commonly understood to be any network that uses the regular ICANN address registration system, but that doesn't mean that the systems all have to be connected together. So if I have a single stand-alone network of my own, and choose to use addresses obtained from ICANN, then I am officially part of the Internet, so if somebody wants me to delete their private information from my systems, they would have the right to demand this.
"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.
