AFAIK the internet was growing fast already in the 1980´s in academia. E.g. RFC793 which defines TCP is dated september 1981.

Also, there was a network community before the internet. Cliff Stoll’s The cuckoo’s nest, describing events from 1986, mostly happens on tymnet. There’s a paragraph in there comparing tymnet with internet.

You might be right about the www. I remember receiving my first chernobyl cows by email. The link above has only ASCII cows, the chernobyl aspect came from embedding ANSI escape sequences in there to colorize or blink the cow. The first mail programs were not escape resistant, so the cows could leak into the gui and even mess with your keyboard.

It might have started in 1989 - but in 1993 when I encountered it it was still alive.

From my terminal, just now:

  ~$ cowsay "This meme hasn't died yet..."
   ______________________________
  < This meme hasn't died yet... >
   ------------------------------
          \   ^__^
           \  (oo)\_______
              (__)\       )\/\
                  ||----w |
                  ||     ||