The living ontology uses the bald term web to mean any hypertext database organized as linked pages, most typically wiki pages or static HTML.

Technically it means the content, links, and weak ontology implied by the naming conventions and core glossary used, e.g. in header tags and footer tags.

The older term hypertext or hypermedia are now mostly found in academic literature and basically have no meaning other than to add stable URIs to other media.

web 2 ?

There is a web trade devoted to optimizing such structures, link ethics and analysis of them. The web 2.0 technologies are most effectively deployed by those with OP-recognized credentials in that trade.

Reviving Advanced Hypertext'', adding semantic web features defined in research systems of the 1970s to 1990s, is advocated by the most advanced practioners. Emergence of a semantic web and eventually sociosemantic web with advanced features like epistemic filters seems likely in the next generation, which some refer to simply as web3.

other terms: website, the web, public web, www

The term:website is problematic as it implies both a sometimes-dangerous spatial metaphor, and also implies that the information is current and actively updated. A web might be simply an archived copy of a prior web service that is no longer operating with obsolete content. Services like archive.org let you search this content even if withdrawn.

It's now common to use the term the web or the public web to mean the World Wide Web. This will probably spread as federated webs obeying common conventions and federated identity methods spread, especially through the use of extranets shared by multiple organizations.

When abbreviated "www", and added to a domain name (spoken "dubya dubya dubya"), it is increasingly deprecated as a waste of airtime, and possibly also for publicizing George W. Bush via his nickname. Some argue that since political advertising relies on repetition, to repeat Dubya's nickname three times before stating any web service name is free positive publicity/votes for Bush.