I think this is a common misunderstanding. The US seems to use "credit card" to refer to all payment cards, but is rarely limit to just _credit_ cards.

I think the distinction between credit and debit cards is more commonly made in Europe.

I mostly mean that I don't have a card that can be used for payment over the usual networks like MasterCard (unless those “Maestro”-cards do work, which I doubt), Visa or AmEx …

IME Maestro cards generally don't work, but there are MasterCard debit cards that do.

Depending on the merchant and the bank, debit cards might not work. For example, the debit card from my bank will not work if used to pay at Amazon (allegedly the bank is too strict about payment details and Amazon doesn't provide something). I had to get a credit card specifically because of this (the debit card works everywhere else I tried it).

My debit card from Wells Fargo is referred to as an ATM card on the card itself. Using my PIN was required and even then some merchants would error. Since the switch to chips in America I haven't had an issue with the card being declined except when a merchant requires me to fall back to reading the magnetic strip.