The first wave was cgi-bin based fork() and exec() a process.
Followed by what you're describing...things like mod_perl and mod_php, where the app code was running inside the apache process.
Then, later, a separate daemon that kept some pool of processes running for apache to open up connections to, like fastcgi.
(We were also running the Netscape web server, because we depended on a plugin for it to insert inventory from our NetGravity ad server. And boy now I feel old.)
At the time, most stuff worked like that from inetd. It would listen on lots of ports and fork off the right program for the incoming connection, then that process would exit when it was done. Long running daemons and lots of processes weren't as common.
The common pattern prior to that was people logging in, reading already synced newsgroups, editing stuff, compiling, logging off, etc. Transient processes that went away.
Whoa. This is what made me feel old for the first time, ha!