OMG I just found the error, 5 min. after posting this. But I’ll leave it here for any poor crusader after me:
The problem was, that my CSV had kind of a “headline” for the first line, as you can see in the original post. The importer visually ignores that and seems to understand it, as it shows the second line as column titles as you’d expect. You never get an error message or any visual hint that this is a problem.
The CSV’s first line must be either the first item or the column names.