I Worked a bit more on my problem, so far without success. Here is what I did.
I found some docs
about the Syntax. It is based on BSD but I assumed no differences.
Based on this I cleared wpa_supplicant from all non essential lines. The default for most parameters are less specific compared to the example. So I assumed that by removing them the settings would allow a wider range of configurations. Also the defaults seem fit for my router. All that remained within the network statement were ssid and psk. It didn't help.
Then I tried the plain key. This also did not help.
I checked and double checked for syntax errors. I also kept an eye on my router to see if any new devices showed up.
My plan is now to configure all parameters explicitly. If that does not work too I will create a serial connection. From the terminal I want to start the wireless connection, hoping to get some feedback from the system. What commend should I use to start a wireless connection?
Alternatively I could test the wpa_supplicant and my dongle from Ubuntu. This might be quicker.