Over time I have gained more insight in the true core of my problem and thus I want to report it again:
My setup is now fully dual, so that finally I could understand the problem.
I have my external HDMI monitor set as my first screen and the notebook as my second screen. After rebooting I find this order reversed. To get back what I want I open system settings - displays and there I see that the HDMI is the first monitor, but this is not true. So I have to switch the primary display to laptop-screen, accept this and then I have to switch the primary display to external HDMI and accept this. The moment I accept the latter I get the two screens switched.
So if you imagine to find out this problem with only one screen active you can imagine why I was fooled so much for quite a while.
My setup is now fully dual, so that finally I could understand the problem.
I have my external HDMI monitor set as my first screen and the notebook as my second screen. After rebooting I find this order reversed. To get back what I want I open system settings - displays and there I see that the HDMI is the first monitor, but this is not true. So I have to switch the primary display to laptop-screen, accept this and then I have to switch the primary display to external HDMI and accept this. The moment I accept the latter I get the two screens switched.
So if you imagine to find out this problem with only one screen active you can imagine why I was fooled so much for quite a while.