The law says if there is a fraudulent charge on your credit card, you are not responsible. I believe the law allows for a variance on debit cards, that I believe is contractually, around $50 that I could be responsible for (if you don't watch, time limit sort of thing).

I've had issues on both cards. My CC has had fraudulent charges two or three times from memory, the last of which was on a card, which had not been activated and strongly leads me to believe it was an internal theft at the issuing company. My debit card info, was stolen/used just once. I had made a LARGE equipment purchase (large enough that I called the bank, transferred money and had them temporarily raise/lift the daily limit), and did it over the phone. A few days later, some company called Lyft, in another state, emailed me one transaction (very short drive at a cost of $15), and then again a few minutes later (longer drive at $300). I contacted their fraud division after my CU rejected the second charge, and I had to tell them about the first charge (which they reversed).
My debit card has never been my primary account. It was only set up due to Paypal requirements and getting some used equipment. I do not know how to use an ATM/never have, and do NOT use it local. (know all too well from LEO relative about skimmers and prevalence) I keep very little in it, and don't connect it with the associated savings (no overdraft protection). Heard too many Paypal horror stories, even though I have never been a seller.

UL, since you seem to be able to actually get a response from DR, ask him who he uses from processing debit card transactions, as I have never found a company that does those only, rather then CC/DC.