We had an experimental web site up in early 1993 (a colleague has email correspondence to Tim Berners-Lee dating back to Jan. 1993 -- mine is long since deleted). But, as many did, when we compared the nascent web to gopher at that time the web came up short (why go through all the pain for only a slight added benefit?) so we eventually pulled the plug on our web server and went with gopher. Boy, am I glad heads didn't roll over that decision!
Those of you who weren't there at the time need to remember that the linemode browser was as good as it got. Mosaic was, at first, a gleam in NCSA's eye, then was only on the Unix platform for a while, then finally made it to the PC/Mac world in the summer of 1993 in a primitive form (JPEGS were not supported inline, only GIFs, for example). Mosaic single-handedly "made" the web. That is, without a graphical client (that could also be a gopher and ftp client), the web was toast compared to gopher. With it, gopher was toast. But by that time most of us had made a serious investment (in terms of staff buy-in, training, etc.) in gopher. It then took us a while to completely retool, convince our colleagues that we had only their best interests in mind when we led them down a dead-end path, you get the drift. Sometimes it doesn't pay to be an early adopter.
So...to get back to the original question. I would bet money that our experimental server was one of the first ten library servers in the world. It wasn't, however, a production server. And as far as that goes, we were probably late to the table, having heavily invested in gopher. We were "live" on the web sometime in March or April of 1994, but we were not "public" (announced) until August of that year. Am I proud of any of this? Not particularly. It mainly shows stumbling steps toward a future we could barely even imagine, and certainly couldn't predict (see above). But if you listen to the interview with Tim B-L, at the URL that Peter Scott posted recently, you'll realize that he was doing the same. Progress isn't pretty or predictable. And only hindsight sees the obvious.