KiCad 9.0.7
Probably due to my missing experience I clicked something in the beginning of creating a schematic which led to the fact that all symbols show their description text (e.g. “Resistor”) in the schematic. Where can I switch off this behaviour by one strike?
Could you please be a bit more precise? Where can this be found? Schematic App? I’m using macOS, FWIW.
You are asking about schematics so it would be a fair assumption that it’s in the Schematic Editor . . . under Edit, look down 
Sorry, I don’t see how to influence visibility of the description field of symbols here.
Try:
Schematic Editor / View / Show Hidden Fields.
“Show hidden fields” was not activated. Also toggling it didn’t have any influence on Description Resistor being displayed at some resistors.
Maybe I had enabled this with the first resistor I created and then it got propagated with every copy I placed. Meanwhile I removed the property manually.