Maybe we understand the terms differently. The normal thing I've seen is that the bank gives you a debit card that you can use at that bank's ATMs. In addition to that, the same card is usually associated with another system, such as Visa, Mastercard, Maestro, etc.
If the card doesn't have any associated system, it can't be used to buy anywhere and can only be used at ATMs, and only at that bank's ATMs. It's very rare (I personally don't remember any, but I'm sure they have to exist) to find one that doesn't have any associated banking system. Maybe in the EU it's different, I'll tell you how it works in the countries where I've lived, I'm not saying it works the same all over the world.
My first debit card was from a local bank in my country, in the mid-90s, that is, 30 years ago. In addition to working at my bank's ATMs, the card was associated with the "Maestro" system and I could even use it in the USA, as long as the ATM or store accepted the Maestro system.
The stores I've seen (in several countries), don't tell you if they accept credit or debit cards, but rather they usually have a sign with a bunch of icons for each system they accept.
In recent years, it seems like all the systems are integrated and people tend to refer to them as debit or credit cards or not even that, often simply asking if the store accepts "cards", referring to whether they have one of the machines that allows you to pay with some type of card or if they only accept cash. This is not the correct use of the term, but it is how I have seen people use the term these days.