Most of you guys are describing what I remember. Thick paper cards and metal cards sounds like what I recall in the '50's. American Express, Diner's Club. I remember my parents had a specific charge card they used for air travel, I just don't recall the name. It was accepted by any of the airlines. Growing up in Atlanta, my parents rarely had to dig for a card, at least when I was young. Most places we did business with just asked them to sign for things "on account". Rich's Department Store, Regenstein's, Buckhead Men's Shop, Glass' Gulf Station, Matthew's Market, Wender and Robert's Pharmacy, Happy Herman's, Henri's Bakery, Harry Baron's, Progressive Club,many different restaurants, etc. At one time I think my mother must have had a credit card for every store in Atlanta, much to my father's displeasure. I never abused the privilege so thanks to my dad I was still able to sign for things years after my folks retired to Florida. Those were the days. I don't think many places in Atlanta still do that. Most of the places I recall are long gone.
The same was true about our family members in the NYC area, so in spite of it's much larger size, the businesses in those neighborhoods also kept "accounts" for people to sign off on. I didn't seriously carry more than a card or two until the late 70's. Like many people, my first card was Sears.