I have a Jap Ps2 model 10000 unlucky for me being in the UK.There is no where to send it for repair not worth it(Brought it used anyway).But mine has a similiar problem.My machine can see DVD films no problem it just seems to have problems seeing PS2 DVD games.Either Sonys DVD game pressing plant is not producing DVDs to the correct spec or the machine has just been setup from the factory incorrectly.I am just Lucky that after a couple of resets in will play DVD games.
I have found that if i put a DVD disk(film) in while in the browser and then change it to a DVD game disk this seems to reduce the problem.As if it doesnt have to re-focus on the disk.As if its set at the right focus from the film.This might help some other Unlucky Ps2 owners.
Give it a try