In short: I confirm the solution worked! 
Here are the steps (from the beginning):
- create bootable Ubuntu LTS USB Stick
- UEFI boot the machine from the USB stick
- “try” Ubuntu (live from USB stick)
- download haos_generic-x86-64-10.5.img.xz
- download USBimager & extract from zip file
- delete everything from the MMC disk with the disk utility
- create one big Ext4 file system partion with disk utility
- move haos_generic-x86-64-10.5.img.xz to /tmp (for finding it easier later as sudo)
- sudo usbimager in a terminal window (otherwise it didn’t let me flash the disk)
- flash the mmc disk
- Reboot the Mini-PC and remove the USB stick
HAOS started right away
Additional steps:
- Instead of starting with a vanilly system I uploaded a most recent backup file and rebooted again
- Logged in - everything was working at first look!
Well, not everything. My add-ons didn’t start at the beginning. I had to reboot the host another time, now everything seems to be fine.
@nukem1232, @keydi, @k8gg, @everyone, thanks a lot for your support, and your dedication!
@nukem1232, apparently I lost the bet, a beer on me! 
Whoever might need support: happy to help!
Damos