Historically, these have been issued by "consumer credit" specialized banks like Sofinco; and retail chains ("carte Aurore"); traditional banks would seldom advertise them, if offered at all.

Things have been changing a bit in recent years. Since the "debit" and "credit" nature of the card is now written on them, French folks have started to request "credit" ones for travelling (to rent a car for instance).

My understanding is that for car rental purposes, anything using Visa/MC (and not a national debit network like Visa Debit in the US) will work, it doesn't actually need to be backed by a revolving credit. At a US gas pump, a Frenchie needs to select "credit" even though the card has "debit" written on it. Still, should the clerk refuse the card because it reads "debit" without running it... better have this "credit"-labeled one.