You can use both PSX and PS2 memory cards in the PS2 but you can save only PS2 games on the PS2 memory card and PSX games on the PSX memory. You better mark which one is which if not marked already, or you'll get frustrated and confused like I did once trying to save my darn game.
Might I add I also had a problem with a few off-brand PSX memory cards that didn't seem to work so well, sometimes it doesn't pay to be cheap.