Look at the surface of the disc's in question. If after the grinding noise happens you can see little hair thin circular scratches appearing on the surface of the disc it's possible that the laser assembly caddy is misaligned. Which can cause the laser eye to physically hit and scrape the disc.
Laser resistance being wrong can sometimes cause the eye to "lift up" higher than normal and thus scrape the disc.
Does the scrape sound appear during "spin up", or when the laser begins seeking?
PS2 fats are finicky beasts. Especially certain models. Some of them require laser caddy height calibration at multiple angles. For example: if one adjustments is correct all but dvd movies will play, if another is out of alignment dvd movies will play but colored discs wont, or even cd roms will play but not dvd nor ps2 games. All of those adjustments have be dialed in "just right" for all of the various types and surface colors of discs to work.
Eventually over time, and use, those alignments drift out of focus.
Don't put tape on the discs. That's a terrible thing to do. It unbalances them and makes them heavier.
Look:
[image|3204864]
See where I've marked with green. That dial/cog adjusts the "foreground/background" tilt of the entire caddy. Some models only had that one single dial/cog.
See where I marked in red. Some models had separate dial/cogs, one for each of the two rails, independently. So not only adjusting "foreground/background" tilt but since they were separate it also had a diagonal angle impact.
Lastly see the set screw marked in blue. This adjusts the "levelness" of the laser itself.
Whether yours has a single dial or the dual dial set up in either case ALL of these factors (including laser pot resistance) all have to be calibrated "properly" in order for all of the various disc types and surface colors to play correctly.
Mark their default positions with something like a sharpie so you can easily return to neutral and start over if needed. Adjust and retest until all disc types function correctly.