AP Style Alert: Don't Capitalize Internet and Web Anymore (poynter.org)
Saturday the Associated Press announced they're changing the rules in their influential stylebook: the words "internet" and "web" should no longer be capitalized. "The changes reflect a growing trend toward lowercasing both words," their standards editor told Poynter.org, pointing out that both words "have become generic terms." Words tend to be lowercased as their usage becomes more common, and Poynter.org points out that "In 2011, e-mail became email... in 2010, Web site became website." In 2013 the AP even revised their usage of the term "illegal immigration," advising "use illegal only to refer to an action, not a person: illegal immigration, but not illegal immigrant," as part of a push towards'ridding the Stylebook of labels."
There are many possible "internets", networks of interlinked networks. The "Internet" is one of them, by far the largest. That is, "Internet" is a proper noun, in the same way that "Bill", "Associated Press", and "Monica" are.
All language is just a collection of common usage. It follows its own evolutionary process. What was correct in the past or present will not, and should not, necessarily be correct in the future. The point of language is to attempt clear communication between people. Correct/incorrect usage should be derived from how it's used by the populace, not forced upon the populace.
It's not about correctness. It's about meaning. The internet isn't a proper noun any more. You wouldn't capitalize the "Phone System", would you? It just isn't a formal name. You wouldn't capitalize the "Interstate Highway System".
And the "Web" is rarely used any more at all. When something becomes that ubiquitous, it doesn't need formality.
You are welcome on my lawn.