For what it's worth, what you have is a Visa Debit (or Mastercard Debit) card, though the branding on them isn't consistent. Debit cards that can't interop with the Visa/Mastercard networks do exist, I have one .. mostly because I'm lazy and haven't asked for a change.
Dunno how rare they are, I suspect rare, but I can't find any numbers with a quick search.