When speaking about the internet : no difference. When speaking in general, website can only mean a website, but a site can sometimes mean a place or a touristic area
because Internet is a noun. and you know, nouns in German always written with a capital letter
As for English, it's because Internet is a name of a particular network (like Earth is a name of a particular planet).
The only indefinite article offered was 'a', but the English translation should be 'He wants AN internet page'.
It seems a verb is is missing at the like "haben" or "bekommen". Is this an often expression?
"Wollen" means "to want". There is no need for another verb in this case.
Technically, there is no such thing as Internet page. But people use the term all the time so why not.
True. People often get the Internet and the Web confused. In French, I hear "site internet" instead of "site web" a lot
Why not? The Web is a part of the Internet, so web pages are Internet pages as well.
True, but not the reverse.
One could have an "Internet Site" that is NOT web-based. Before about 1995 or so, all Internet sites were non-web.
Thus, »Er will eine Internetseite, aber keine Webseite.«
Alternatively, one could assume that "Page" implies HTTP. But I don't.