I would favor speaking of "the internet" versus "internet," or even "Internet." If the difference between Internet and internet is the degree of connectivity between particular sets of computers, it is far too ambiguous for consistency.

The reason I favor the term "the internet" is because it can be consistent amongst many people versus attempting to make a distinction between Internet or "the" or "an" Internet or Internets when the only difference is purported to be some ratio of total computers on Earth to currently connected computers. If we were to say that the word, Internet, was to speak about any subset of computers, then it wouldn't be a proper noun, as many "Internets" would exist at any given time. If we were to reference the internet as only one entity, it seems correct to identify it as, "Internet." The problem with that is that "The Internet" is never sufficiently cohesive as to give it its singular.

As is frequently typified in "The Colbert Report," the use of the word "Internets" is frowned upon in contemporary speech. Additional word breaking or hyphenating, such as inter-net or inter net, is also seen as something alien. The only permutation that seems reasonable is "the internet" which is already plural, and not a proper noun.