A bank card is a type of card. Credit cards and debit cards are both bank cards. Prepaid cards are another type.

With any type of bank card, there's a bank that guarantees to a merchant that they will later receive a payment. With a debit card, the guarantee is backed by money you have on deposit. With a credit card, it's backed by the bank's money, which is higher risk for the bank.

Two US companies, VISA and Mastercard, have big networks for processing transactions with bank cards. These networks act as intermediaries to connect merchants (who want to accept payments) and banks (who issue cards) together. It's much simpler for a merchant to send a request to (say) VISA than to figure out which bank issued each customer's card. The payment networks also define, publish, and enforce standards and rules for the payment process.

These networks aren't banks. But they are, in a sense, bank card companies because they are part of the bank card system.

So in other words, European consumers have an account at a European bank that issues them a card they can use for purchases at European businesses, but US networks connect it all together.

US “bank cards” as in US payment processors such as Visa, not “US bank” cards.

Visa and Mastercard are both US companies and they issue both debit and credit cards. If you have a UK or EU bank account with a Visa or Mastercard, regardless of currency or whether it's debit or credit, you are still ultimately reliant on US companies to clear transactions every time you use it. That's what the EU want to reduce.