memory card for playstation 2 question:

I have been using a memory card for my playstation 2 and I saved data for some playstation 2 games.

However when I played a ps1 game on it, it said memory card has been removed, and it won't save for my ps1 game.

Is there a separate memory card I have to buy to save the ps1 game?

I can't figure out why the memory card works for ps2 games but not a ps1 game.

Also, when I use it to play ps1 and ps2 games I use a flatscreen t.v. and it kinda looks like crap , is there anyway to make the resolution better? When I used to play it on a crt t.v. it looked fine though. Obviously.