When was the term "Web site" (or "website") first used?

I think 1994 would be about right. I remember the Mosaic browser coming out in 1993 and then in 1994 Netscape navigator came out, along with their own web servers, and suddenly the public could explore the web. In wikipedia it says:

Tim Berners-Lee is credited with inventing the World Wide Web in 1989 and developing in 1990 both the first web server, and the first web browser, called WorldWideWeb

So the word "web" was already floating around along with "web server" since Tim created them.

Some of the first hosts to offer web site hosting back then where Tripod and Pipeline. They were the first ones I used once I left AOL's and CompuServe's closed networks for the wider web. I would dig more into early hosting companies as they would have been the first to advertise in the magazines and coin the "web site" term.

Also take a look into "List of websites founded before 1995" on Wikipedia. "Website of the day" site is listed as up in 1994 so I think that is a safe bet. Navigator introduced images and new features that suddenly made web pages much more interesting. That's when I moved into web site development. Come to think of it even back then it was web site development so the term could have come from Usenet or CompuServe/AOL which would explain why there is no obvious starting point. Usenet was the Reddit of the day and lots of stuff was argued and talked about there that trickled out to the wider web.

I wish I could pinpoint it better but I thought I would share a few thing that might help you narrow down your search.

The first written usage, according to OED Online, is from Jan. 25, 1993. It is from FreeLore Whitepaper Repost in alt.uu.future (Usenet newsgroup).

World Wide Web sites: info.cern.ch (128.141.201.74)..[etc.].

You can find the full text on this site also:
http://wiretap.area.com/Gopher/alt.etext/014.freelore

Note: It is two months before it appeared in Computer Shopper magazine (Mar., 1993 issue), according to OED.

I started building websites in early 1996. One of my first websites was a called napalinks.com. For many years it resided on the first page of Google as a portal to all visitor websites located in Napa Valley. My website worked because I let visitors in and then released them quickly as soon as they found a link to the website they were looking for. I explained my concept to other programmers was told that this was not the purpose of a website. The purpose was to keep traffic inside the website, not to release visitors out of it. I was told this is why it is called a website; the creator of the website is like a spider keeping his prey inside of the web. Letting the visitor out defeats the whole purpose of creating the site in the first place.