The terms "Internet" and "World Wide Web" are so easily confused by John Q. Public that it would simply make it worse to use both terms, although I understand your thoughts about the difference.
Most "intraweb" sites are rather boring compared to the wealth of information on other servers --- after all, the "intraweb" is just one company, organization, person, whatever. Isn't that why the "World Wide Web" caught on; people who were using early browser software to look at documents on their local networks discovered that some authors had included links to documents on distant servers, then that trend caught on with more server admins/authors, pretty soon you could not only read your research and the research of your colleagues, but also that of New Zealand Polytechnic, the Sorbonne, the University of Southern North Dakota at Hoople, etc., etc...
Then M$ and organized crime discovered it, and the rest is sordid history....
Stolzy's got it; you can call it that, but it's not likely to catch on....