there is a red 5V wire and a black ground wire they both go to a male plug that goes into the female plug shown on the left in that picture.
it appears they have torn the female socket and trace's and why the wire's are soldered to the pcb's via's if that is your ps2 in the picture.
they way it is wired to the PSU will not hurt it either it is just being forced to run at 12V instead of the stock 5.
5V would be more quite but the console would run warmer.
12V the fan should be a little over twice as loud as stock but also run the console cooler.
most pc and game console fans can run 5-12V safely it is just how loud you want to go but if it does not get hot then there is no reason to make it louder really.
I have never had a ps2 overheat but maybe it needed a good cleaning to run cool at 5V and instead they made it 12V thinking it was the fan when it just needed a good cleaning.
but then again they may have had a large hdd in there that pumped out some heat and they made it 12V to prevent it from getting to hot in there
I have a fan pulled from a new console that did not power for $4-5 bucks shipped if you want to use a stock fan and try that molex at 5V the wiring to the PCB looks good but you may want to check the wire to the female plug if you cannot solder good before you get another fan.