What matters are the legal and contractual rights and obligations you have against the card issuer. In the US, these are historically different for credit cards (Fed Regulation Z) and debit cards (Regulation E), but since it's now effectively the same two schemes running it all and imposing their additional liability protections (largely motivated by considerations of brand perception, which would suffer if the same logo sometimes confers weaker protections).

The main practical difference nowadays is that in the case of debit cards, you're out your own money for a few days, while with credit cards, the only thing that temporarily suffers is your open-to-buy/line of credit.