There are some differences behind the scenes.

A week ago I was in this restaurant and I had almost no cash (as usual where I live in Northern EU); the restaurant terminals were rejecting debit cards from the two biggest banks in the country. After several attempts, I just paid with my credit card (which is configured to pay immediately from the account, not at end of month) and that worked.

That's probably because a credit card can go through with a transaction even if some of the involved parties fails to respond.

I think in the US, that type of card is referred to as a 'check card', and is basically a bank debit card with the additional ability to run it as a credit network (usually Visa) transaction with signature instead of PIN entry. Is that similar to what you have?

I am not sure, they both operate with a chip and pin code. I think it really is just a difference on how the banks settle the transactions between themselves.