My issue is now fixed
Unfortunately, I still don't know what happened
This is what I did, for future reference:
Started from the broken 1.6.0.9 version
rolled back to 1.6.0.8 in the 1-click upgrade module
cleaned all caches and looked at the front office. Everything was ok now.
Updated once again to version 1.6.0.9. In the process the upgrade process upgraded 100 modules as well
Cleaned all caches and looked at the front office. Everything was still ok
Looked at the modules section to see if there was other modules to upgrade but there where no new upgrades.
Let's hope this is the correct way to proceed in case something like this will happen again.
