Talking to a store clerk over the phone:
Will I be able to use my credit card for the gift card?
Would I be able to use my credit for the gift card?
Are both okay here? What's the difference?
I'd definitely use 'my credit card'.
Neither is wrong, though "my credit" sounds slightly awkward to me: "credit" would work on its own.
There is a difference in the two phrases, though. While "credit" in AmE is usually short for a "credit card," (one backed by VISA, AmEx, or similar), there are some stores that still issue credit. If I was working in one of those stores and was asked if "credit" could be applied to a purchase (rather than a "credit card"), I would assume they meant store credit and check the records to see if this customer had an account.
[Not a teacher, just very well-read.]
I disagree. In AmE "credit card" is quite common whether it be a major credit card (Visa, Master Card, Amex, etc.) or a store credit card. "My credit card" is entirely natural.