Re: Coming to a wallet near you...
"That's a "store card". Loads of places do them.
Basically it's an in-house HP agreement, but you only have to do the paperwork once up front."
It passes the "duck" test for a credit card. It's a sliver of plastic that I can hand over in a shop (ok, only one brand of shop, which is why you think it's only a store card) to make payments on "instant" pre-approved credit. And I get monthly bills with the option of paying any amount between an arbitrarily-calculated[*] minimum and the full owing balance. That sounds like a credit card to me, even if it can only be used in one brand of shop. Sure, it isn't a general-purpose credit card, usable at almost any outlet anywhere, but it is still a credit card.
Another point to bear in mind is that some store cards are charge cards, like the classic green/gold/platinum Amex cards, where you have to clear the balance every month. The Sears card I had wasn't like that.
[*] Yes, the calculation of the minimum payment is arbitrary, in the sense that there is no particular virtue in the percentage-of-full being any specific value so the card issuer selects an interesting value, subject to any consumer credit rules and/or Visa/MC network rules. Making the value higher tends to decrease the utility of the card - the largest balance limit I can possibly afford is determined by the minimum-payment fraction and my ability to afford a monthly payment. (In reality, a sane person would say he could only afford X amount less than that, in order to have some level of margin, but we all know that sanity is a bit scarce when we are talking about consumer credit.) Making the value lower attracts more of the more reckless spenders that we, in an ideal world, would not want our bank giving credit to, and if the value is low enough, the cardholder can get stuck with an unreasonably long repayment time if he stops buying with the card but doesn't pay more than the minimum...