Update:
I reinstalled the nouveau drivers and rebooted, which did not help. Then I reinstalled the NVIDIA proprietary drivers and rebooted. Ironically, I now get the “NVIDIA driver missing, falling back to nouveau” message and problems all seem to be resolved (external screen and brightness control work, the “Settings/About” section shows I am on Wayland and I am using NV166 graphics).
(I have not yet checked if the third entry upon login appears, will do that later).
Nonetheless, I’d be interested in knowing why this happened and how I could prevent it in the future.