@Anne: different story for you. Switzerland is not a member of EU and has added a step that most EU countries haven’t. You need to apply for a Swiss certificate in advance, so they have a chance to evaluate your vaccination credentials before you arrive. It is done online, and you need to upload pdf copies of your CDC card, passport and proof of stay (e.g., hotel confirmation,) pay 30 CHF and submit. Within five days (usually less), if approved, you will get a pdf file with a QR code that you show to get into Switzerland and to get into any indoor venues. So that would be your Swiss Covid certificate. A similar process is/was required for France, at least when Chris came at Christmas. I’m not up to speed on French entry requirements if they have changed.
Chris’ question was how to convert an EU certificate to a Swiss certificate, as he’s already been through that process in France. My answer is that he does not have to apply for a Swiss certificate, he can convert the French one he has at no charge.
Outside of Switzerland and maybe France, most EU countries accept the CDC card as proof of vaccination for Americans. This is why it’s so difficult to keep on top of changing requirements - each country can set its own rules, not EU or Schengen countries as a bloc.
P.S. those are the facts - my opinion is that this process is actually good for you, not just a PITA. If you have the QR code, you won’t have any restaurant owners or security guards squinting at your CDC card many times a day, trying to understand if it’s valid or not. You will have the standard QR code, they have the standard QR code reader, beep and you’re in. It simplifies your life and theirs, and should reduce the stress of your visit.