I’ve cleaned Nominatim’s database. The district results are fine now. But the place nodes are still used as label in multiple relations at admin level 9 and 10.
Example for Žilina: admin_level=9 and admin_level=10
From afar I’d say the place member in level 10 needs to be deleted because the relation describes a suburb, not the whole city, but this is something the local community has to sort out.
Note that I consider Nominatim’s behaviour correct here with respect to the current tagging: if a admin boundary has a place node as a member with role ‘label’, it inherits the information from the place node because the ‘label’ role indicates that the place node is simply a different representation of the same object (area vs. centroid of the town). If the same place node is in two relations, both inherit the same information.