Clearly from my answer above, I believe the Internet is a proper noun, the name for a specific collection of networks. That’s also why the definite article is used. With a small i, and internet is a common noun, a type of network.
This is also why we generally do not see “intranet” capitalized. It, too, is a common noun, a type of network (one limited to within the organization). Even when people use the definite article (e.g. “You can find it on the intranet”). It is clear from context which intranet is meant. But there are many intranets and everyone knows that their organization’s intranet is not the “one and only”.
Not so with the Internet. While there are many networks that run TCP/IP, there is only one Internet.