internet vs Internet
To non-techies, "internet" already means the global internet. "Capitalize Internet if it's the global internet" is irrelevant pedantism of the highest order. Every business tech article I've read refers to a company's mini-internet as an "intranet," and home networking enthusiasts just call their mini-internet the "network" (e.g. "What's the password to your network?"). (And if you want to go into what non-techies call it: to millions of people, the "internet" is a disc and a plastic box given to them by their ISP. And when something goes wrong, the first thing they say is "the internet is down." I doubt they remember to capitalize it when appropriate.)
To most everyone, "internet" is a thing, not a title.
In this sense, "Internet" is a proper noun, used to refer to one thing rather than a general class of things, and proper nouns are generally capitalized.
I've always understood it to be that Internet = global network, and intranet = local network.
"Internet," without the capitalization, is also a common noun. It is short for "internetwork," which means a network of intranets. What we now call the Internet is an internetwork and used to be called "ARPAnet."