Not an expert, but that noise is the laser pick-up arm reaching the end of the "slot" as it's having trouble getting focus/reading the tracks.
You could try disassemble your PS2 and clean the lens for dust and/or fine tuning a screw (turning a quarter of a turn clockwise or anti-clockwise) on the laser arm to regulate focus (you should still find
tutorials online), but opening a Fat PS2 is somewhat a tricky operation because of the "delicate" power/reset button flat cable (you'll understand if you have opened one).
If I would have to guess, I would simply say that the laser is old and nearing it's EOL.
If you want my opinion, as you have a Fat PS2, get hold of a network adapter and a 3.5" PATA HD, or better an SD card to IDE 2.5" 44 pin male adapter plus a 2.5" to 3.5" pin adapter — yay, no moving parts! — the only drawback is the console 2 cm. fan screaming like a jet engine on take-off. Also look for OPL to use it with your network adapter...
There's also the option to use the network part of the network adapter, but I prefer the internal disk even if I have a PC NAS running 24/7 tho... ^__^