paying with a credit card in physical stores is rare in Europe (and usually considered very strange).There is no difference between the mechanics of paying with credit card and paying with a debit card, at least none that I've ever noticed. I hold up my card to the reader and sometimes enter my PIN. I'm not even sure if there is a way for the person accepting the payment to tell if it's a credit or debit card I'm using.
I've seen Americans try to pay with a credit card in a supermarket in the Netherlands and have it fail.
I've had places fail to accept my Swedish debit card in the Netherlands. It's more to do with some places only accepting local cards or not accepting, for example, Master Card, rather than debit vs credit cards.