The European (I assume, I only know the ones from Germany) debit cards are distinctly different from credit cards or the debit cards you described in that they
* don't have the credit card like information like card number, verification number or CVC code.
* They are directly linked to a bank account like you described.
* They have the account number (IBAN) printed on them. Knowledge of that number doesn't get you closer to the money in the account, though.
* To actually use them on an ATM or at a POS, you usually use chip and pin. In the POS case, the POS randomly chooses between PIN and your hand written signature as proof of ownership.
* You can't directly pay with them online, though (ignoring extra features some variants have).
* To pay online if you only have one of those cards, you usually pay by wire transfer using your account number, the pin number from above and a tan (a single use number created for this transaction only). Money reaches the recipient between a few minutes and a day later.
* Practically everyone with a bank account has such a card.