You mention an interesting thing I didn't realize before. Carte Bleues, the brand, seems to have been sold to Visa but CB, the payment system would still be a separate thing and now mean Carte Bancaire. It is not helped by the fact that french people literally adopted carte bleue as a generic name for payment cards.
Thanks for the info. By the way, is there any difference, apart from not having to rely on foreign parties, for the buyer between selecting CB or Visa/MC for his payment? And I suppose it's the shop owner who decides the default when the buyer doesn't choose.
https://en.wikipedia.org/wiki/CB_Bank_Card_Group