Hi,
I've checked again your link and there is another bigger problem.
If I click on your link (
www.poterie-ceramique-art-de-la-table.co...shop_payment_methods
) I arrive on the checkout with already a product in the cart, even though I didn't add anything to the cart yet.
Normally, I should be redirected back to the products listing as my cart should be empty.
I've tried on different browsers and observed the same issue.
This indicates that you must have some agressive caching on your pages and because of that, everyone sees the same thing on each page. But that shouldn't be the case. I shouldn't be able to see the cart of someone else. Similarily, that also means that the issue with the preselection of the payment method comes from the same cause and it's actually the selection someone else did and I see it because of the cache.
In fact, if I change the selection of the payment method and refresh the page, I still see the old selection, and if I try to click on the next button, I get an invalid token error indicating that the token generated when the page was generated has already expired.
So all of that points to a caching issue on your website.
Please deactivate all the caching on your website and web server and it should help with all of these issues, including the selection of the payment method.