In the US a debit card is free and standard for any checking or savings account.
It's functionally the same, yes, in the sense they are just plastic cards that perform transactions, but as I mentioned in my post there's a big difference in the financial impact. I won't pretend to know the rules, regulations, etc in Europe for credit cards, but when I traveled there I used my cards for everything and only had a problem once with CaptaineTrain booking a train ticket.
With one of my cards I get 3% cash back on dining. So while I'm traveling throughout Europe I'm getting 3% off every meal. If you're paying in cash you're still paying full price. Soooo why pay in cash and not get a 3% discount.
I've also noticed recently that Yelp is offering discounts when people link their credit card in their Yelp account. I saw something like 7% off of your meal for doing so at one restaurant. Now I haven't done that yet, I'm still considering the privacy implications, but it is there.
In the US at least, and I suspect Europe, you should use a credit card for every single purchase.