A card case typically refers to a card wallet that folds, whereas a card holder (aka minimalist wallet) does not.

For that many cards, I'd definitely the recommend the folding variety, as it's easier to get your cards out.

If you do go for a cardholder, I'd recommend finding one where the middle pocket is open on both the top and one of the sides.

More cards can cause tension, making it a bit difficult to get cards in and out otherwise. If you don't use the center pocket much, it's less of an issue, but the recommendation still stands.