Cards have been used by the banking industry for identification purposes since the 1940's chiefly in the USA. Some banks had a system of small loans where cards were issued to borrowers with a good name with the bank. This came to be more sophisticated and formalized in the credit rating system. Cards with "Time payment plan" or "Time Plan" were issued to such short term borrowers.
Many USA banks began trials of Credit cards in the 1950's, and most of these aligned themselves with either Master Charge, or Bank Americard, known today as Mastercard and Visa.
From the late 1960's and early 1970's banks began to trial ATM technology and EFTPOS, and formed business arrangements with the telephone companies to put these ONLINE.
In this section we have a few articles on these systems. Enjoy.
Credit cardsATM cardsEFTPOS

The world's first EFTPOS trials in France using Chip card technology has an article under Chip cards section:
IPSO Trials