The first question is always: is this information in OpenStreetMap.
For the name of the island, it probably is.
For the Bayrischer Wald, I doubt it, as this is a region without well defined boundaries.
Then if it is in OSM, you need to know how it is mapped. Rügen is probably not mapped as an administrative region, so it will not show up in the “stack” you get for a given coordinate (afaik).
So I fear that the current implementation of Nominatim cannot return the answers you are looking for.
Perhaps someone else will prove me wrong.
Yes, Rügen is mapped and when you search for Rügen Nominatim will return the location of the island. But when you search for an address, Nominatim only returns administrative entities. It will never return geographic entities like islands, forests, etc.
Thank you! Yeah, I think I need both services . Fast reverse geocoding for locations and overpass for more informations about a region or a island. Can I use the same database for Nominatim and Overpass? i want to install both services on a single server