Since I don't know much about PS2s, I can't say what exactly is wrong, I can tell you the high drain batteries won't make a difference. The advice you found was mostly right, it won't take any more amps than it needs, yes, but if you accidentally short something there is much more power that can come out and pop whatever was shorting
I would suggest looking for a 79001 as they are most common, but there are 79000, 79001, 79002, 79003, 79004, 79006, 79007, and 79008 models. As for voltage, it will be labeled somewhere around 8.5v, but the PS2 will run all the way down to around 6ish volts on stock regulators (as mentioned to me by gman).