[ommadawnyawn]

I did a bit more digging via the lists on my site (which focused on chiptunes but I've started adding streaming and redbook music to) and found a few more, which I posted elsewhere on reddit:
"When it comes to techno, the earliest similar examples I've found are from Dynamix (unreleased C64 game), Misty Blue (track 15 on vgmrips, might also be the first house music with tracks 13-14) and Bionic Commando (C64, stage 3). Solvalou from 1991 features some ambient techno, and of course Streets of Rage is mostly techno and house (see also Dyna Song from Dynablaster/Bomberman, Zone Warrior, Fascination or the theme from Gods, all for Amiga and from the same year).

Captive (AMI, 1990) sounds vaguely industrial but also like 80s electro, rock, and art of noise. Extase (also AMI, 1990) has some elements like that as well, but mixed with new age and "world music" influences plus it has an interactive element when you play it. Killing Game Show / Fatal Rewind (AMI, 1990) sounds kinda like Ministry, so industrial crossover. Some tracks from Alien Storm (ARC/MD, 1990), and the theme from Speedball 2 (AMI/ST, 1990) also have a vaguely industrial vibe about them.

For alternative rock or grunge, it might be skitchin' (1994), wild woody (1995, cd, also features a rap rock track) and comix zone (1995) for chiptunes and original work (there was also a CZ release with a CD OST album that came with it).
Edit: Oh and there's also Quarantine (PC, 1994) which had various lesser known alt rock bands in a licensed CD ost. I guess that's the earliest example for now. Then there's the 1994 version of Road Rash which did the same thing, with bands like Soundgarden and Swervedriver featured, the latter being in part influenced by Pixies and MBV I think. But I wouldn't say these examples had a larger impact on game music as a whole in the 90s.

The 7th Guest (1993, cd) sounds similar to gothic rock at times.

B.Rap Boys (ARC, 1991) seems to be the first hip hop focused OST and also features rap vocals on various tracks, while Ultrabox 6-gou (PCE CD) from the same year features one rap song."

While there are exceptions of games adopting new music pretty quickly and techno for example became increasingly mainstream in the early-mid 90s, I would say it took fairly long for hip hop for example to get much exposure in game music, and indie rock I've only found a few examples of like Moon (PS1). Some genres like industrial are used in specific genres like horror (Silent Hill). But there are many mid-late 90s games I haven't looked into yet.