I found out two posts that explain what could happen.
From the first post, I learned that the settings are saved in files when Plasma closes. That explains why the settings were changed but not kept when I rebooted.
From the second post, I learned that .config/plas,a-org.kde.plasma.desktop-appletsrc contains also settings for the right-click context menu.
Since I don’t have that issue anymore. I take that configuration file got corrupted. When I changed the primary screen, I forced Plasma to write that configuration file. The first time I did that, Plasma could have “crashed” before writing the configuration file.
It also seems that having two monitor of the same brand and model confuses Plasma. I connected a different monitor and these issues are vanished (for now).