Crepello's correct - the machine (BA, anyway) is only looking for confirmation of identity, so any CC, debit card etc will do.
This is only true for Travel Agents' or Third Party Bookings. Bookings made using ba.com require authorisation which the self-service kiosk carries out behind the scenes. For these bookings, if the correct credit card is not used, the passenger will be referred to a check-in agent who, if he/she doesn't see the right card, will need to get the Ticket Sales staff involved!