I used the wrong words. A local card from your local bank is a defacto debit card by nature. One might call it an atm card, but you can buy your groceries with it. And in the EU you can buy your groceries in the whole EU. But it still is a local atm card. Probably not the type of debit card an American thinks about, if you say debit card.
If your card is Visa, it is a Visa card. That it also is in function debit and not credit, is not of any relevance, except for cases where debit visa cards might not get accepted for whatever reasons, maybe higher risk of transactions failing. Visa is not your local bank. Your local bank cooperates with Visa to give you a Visa card.
So I am totally unsure what is meant, if an US based merchant proclaims, that credit and debit cards are accepted. Do they talk about local grocery store atm cards or do they talk about the no delayed payment cards of the major credit card companies?