My experience is it's the other way around. I am the Technology Director at a school, and when a teacher tells me "the Internet is not working," he is saying that when he clicks on the Internet Explorer or Chrome icon, he's not getting what he expects. Rather than the web becoming the Internet, to them, them Internet is only what they see through the browser. The fact that Dropbox or Facetime also uses the Internet does not occur to many people.

Users don't care about networking protocols as long as the Dropbox application is able to synchronize their documents.

Exactly!

The "web" as in Tim Berners-Lee(TBL) "world wide web" is http+html which is "documents".

Saying TBL-web is "best as a document platform" makes perfect sense. It was defined that way therefore, use it that way. It's tautology.

Back in 1993, if we want to say "X is best though of as a network app platform" we'd have to use the word "internet" instead of "web" for "X" to be conceptually pure and technically correct (the best kind of correct.).

But now we have things like Dropbox (and thousands of other "web apps"). Dropbox runs on "http".

Dropbox "syncs". Syncing is an app. Dropbox is an app. Dropbox runs on http. Http is "the web". Apps run on the web. The web is platform for apps. That's the reality of where we are today.

Dropbox can throw away HTTP and replace it with TCP, UDP or any other IP based protocol and it will keep on working.

This is they beauty of the network communication protocols.

Dropbox without using http_port_80 would only work in a laboratory. One big reason Dropbox is valued at $1+ billion dollars is that it uses http so everyone can use it easily. Web browsers didn't require radical changes. Corporate firewall rules didn't have to change. That's how they got quick adoption and millions of users.

A business decision to use a specific network protocol doesn't define what web and network protocols are.

It is is just a consequence of what network protocols happen to be open by default.