A debit card is a credit card that immediatly charges the associated bank account.
The card you got from your bank probably is a card that also does this, but it is not the type of card that is meant by debit card. At least not when you read it on an US merchant site.
Can you pay online with the card at all? Or do you need to make a SEPA money transfer? That will withdraw from your bank account, but it is not the same as "paying with your card", even if the number of your card looks suspicously like your bank account number. If your "card" is ok for purchasing in other countries, did they mean outside the EU or inside? ;-)
You are in the EU, you should be able to use paypal and pay paypal directly with your bank, if you have no international cards.
It would be interesting if someone could explain the difference, or rather why debit and credit cards are mentioned separatly. Is it because on a "real" credit card, the payments can be announced and reserved but not actually made?