I guess that you are located in Europe and that you bought a used memory card form that region? I think I've read somewere that card that has saves from PAL games are sometimes unable to save NTSC games due to different formats or something like that. Or it could be a problem with the console itself, if you are unlucky. I think you should by a new and unused card and use it only on your NTSC console. All new memory cards need to be formatted before you can use them so a brand new card should work.
I guess that you are located in Europe and that you bought a used memory card form that region? I think I've read somewere that card that has saves from PAL games are sometimes unable to save NTSC games due to different formats or something like that. Or it could be a problem with the console itself, if you are unlucky. I think you should by a new and unused card and use it only on your NTSC console. All new memory cards need to be formatted before you can use them so a brand new card should work.
Ok. I will get a new PS1 memory card and see if that helps.
As far as PS2 memory cards go (pretty sure the same holds true for PS1 memory cards), they are all manufactured the same way in every region. This is why you can use a Japanese controller on an NTSC PS2, and a NTSC memory card with imported games.
Try formatting that PS1 memory card you got and see if that changes anything. If not you can go ahead and try the one you linked.
The PS2 Slim made a small change to the memory card pinout to block third-party memory cards. It also breaks some official memory cards that don't work without the extra power pin. You can try other memory cards until you find one that works or get one of the PS2 Slim multitap adapters that has a memory card passthrough. Those usually have all the power pins wired like the original fat PS2.