When I got my first modem in 1995 (A Creative Labs Modem Blaster 14.4kbps), it came with a bunch of shareware and demos on a CD that was meant to be played over dial-up. Stuff like Warcraft 1, Descent, Heretic and Doom. So I installed them on my 486 and then I also installed them on my cousin's computer so we could play via direct dial-up connection. Of course, it would disconnect us anytime someone picked up the phone on either side.
The first game I properly played online, though the Internet, was Starcraft, via Battle.net, with a 56.6k modem.
I feel Starcraft very important in introducing a lot of people to online play, both younger gen Xers like myself as well as older millenials. It came out just as internet penetration started to become more common (1998) and it also offered a streamlined, easy to use platform thanks to Battle.net.