Is there a term which covers ATM cards, credit cards, and debit cards?

The industry itself uses the term Payment Card. See this Wikipedia article for details. In IT you frequently hear the term "PCI compliant" to refer to software that meets security standards set out by the Payment Card Industry Security Standards Council, a consortium of players in the payment card business.

In practice the term isn't used often, but if you're looking for a single term I don't think there's a better one out there. I think the reason most merchants spell out which types of payment cards they take - "Visa, Mastercard, ATM and gift cards" for example - is because there's a lot of choices out there and most places do not accept all of them.

Some places have used the term plastic. It makes it clear it is not cash, or a check. It could be a credit card, debit card, or even a gift card.