Actually, you can use either your Disney Visa card or your Rewards Redemption Card to get the discounts.
Merchandise discount fine print:
Must use your valid Disney Visa Card and/or Disney Rewards Redemption Card as form(s) of payment at the time of purchase for full amount of purchase and mention this offer.
Dining discount fine print:
Must use your valid Disney Visa Card and/or Disney Rewards Redemption Card at time of purchase for full amount of purchase.