BUT... in a more colloquial sense of how "web" might be used retrospectively, as sort of a loose synonym for "online", then absolutely. Going back well into the 1980's, and then into the 1990's, we had all sorts of online services, including discussion forums. There were the big services like Compuserve, The Well, AOL, Delphi, GEnie, Prodigy, etc. And then there were individual small dial-up BBS's[2] by the bushel-load.
At their peak there were many thousands of them in the US alone. There were entire magazines printed and distributed that were effectively nothing but lists of dial-up numbers for BBS's, and magazines like Computer Shopper had big sections dedicated to listing those as well.
And then there was, as you mention, Usenet.
So yeah, even back in 1991, there were plenty of forums that might not have strictly been part of "the web", but that someone in contemporary times might casually describe as a "web forum".
I grew up on AOL 2.5 and the hacking/warez scene there. So many good times with 1IM punters, cchats, scrolling, and phishing OH and INT accounts.
the well has been around since the mid 80s. It was a bbs and then a few other weird things until it became a 'somewhat standard' web forum in the mid 90s -- but if I remember right when it transitioned from BBS there was an effort to move the conversations over to the web forum.
delphi forums too, which similar to the well has been through a few incarnations.