Good morning Carto_Ferret,

building=hospital is appropriate in most cases assuming the amenity=hospital polygon is correct which is not the case.

way/304882202 houses a dance school and alternative medicine cabinet, way/304882202 and way/303662304 are a residential buildings (it is pretty easy to determine it by their roofs since hospitals here are quite old and they almost always have flat roofs). You can also see the party wall (https://en.wikipedia.org/wiki/Party_wall) on the first building, which is not common for hospital buildings. The strangely shaped building might be some service building or it may be some doctor's office. I will leave it as building=yes so it can be surveyed.

I've corrected the hospital polygon and the building tags.

Best wishes,
Dimitar