This is a partial update, since my problem is not yet solved.
I called Apple HelpDesk last Saturday as you indicated (thank you) and I passed 2 hours with them. They collected all the information I could give them. Then I was invited to reinitialise my phone (which is cumbersome, but no other solution than do what they want) and got an appointment for today at 12.00 (Paris time).
Note: this was a partial reinitialisation that did not put the phone back to factory settings (so I did not need a backup to restore it), but I had to reconfigure every app again. Credit cards are also deleted as well as ALL wifi passwords saved on the phone. (solution to it: go to the Passwords app, check the Wifi passwords stored there, save them somewhere else).
Today I had first on the phone a technician who after 30 min passed the call to an ‘upper level technician’. He (they all connect to your phone) did some actions on the phone (just to verify once again the problem and some settings) and collected some info via screen capture as well as the history of the phone. Next appointment is due for next Tuesday.
If you have the same problem with a second iPhone which was also updated to iOS 18, please mention it to them. It is an important information.