When buying online I cannot use the debit card. Otherwise, sometimes I get high transaction costs on some cards instead of others.
To solve this, the EU should forbid banking barriers, and legislate that everything can be bought using any type of credit card at no additional fee for either who buys, or who sells. This makes or breaks the DSM.
You'll be aware that the Commission took some robust measures a few years ago to cap bank transfercharges in the Euro area. There's also quite a track record of competition cases regarding credit card companies, if memory serves.
I've noticed that my debit card doesn't work across some borders consistently, when I'm outside Belgium, even within the Euro area, especially for cash withdrawals. I wonder whether the problem is linked to the absence of agreements between some banks, rather than being a legal problem.
I'll need to involve people who work in finance to get a better answer for you.