This time the 'tape across toner window' trick would not work, even with black tape. The 'toner empty' message would only go away after doing this strange procedure which probably resets some internal state so it re-checks the toner state:
Open the front cover, then press the Back (german:Storno) button (the left bottom one of the four round black buttons), then press * 0 0 and then * 1 0. (Do not press 1 or 2 as indicated in the display. This is for the drum.) After this the 'toner empty' and also the 'toner almost empty' messages were gone and I could perfectly print again.
A subset of * 0 0 and * 1 0 might be sufficient. Not 100% sure the black tape is necessary at all.
Now the printer is printing happily and in perfect quality again. Lets see how long. :-)