Here's my take at a definition: A privately owned network is "on the Internet" in the usual sense if it uses IPv4 (or maybe 6), is properly assigned IP address

Here's my take at a definition: A privately owned network is "on the Internet" in the usual sense if it uses IPv4 (or maybe 6), is properly assigned IP addresses that are not in the private use area (publicly routable IPs), and has some systems that can exchange packets with the diversity of other networks that make up "the Internet". NAT devices tend to complicate this definition. But that's a technical definition. You're asking about how it could be politically regulated. Almost all privately owned networks (other than ISPs themselves) pay an "internet service provider" for "internet services". So you're not going to weasel out of regulation by nitpicking about the definition of "the Internet".