Thanks for replying.
~/Desktop did not contain anything. But I've found what was wrong. Sometimes a night of sleep lies in between the problem and the solution. ~/.config/user-dirs.dirs had to look like this:
XDG_DESKTOP_DIR="$HOME/Desktop"
XDG_DOWNLOAD_DIR="$HOME/Downloads"
XDG_TEMPLATES_DIR="$HOME/Vorlagen"
XDG_DOCUMENTS_DIR="$HOME/Dokumente"
XDG_MUSIC_DIR="$HOME/Musik"
XDG_PICTURES_DIR="$HOME/Bilder"
XDG_VIDEOS_DIR="$HOME/Videos"
I looked at the file in /root as root and noticed the first line showing
XDG_DESKTOP_DIR="$HOME/"
I start to learn about *.desktop files, too, which I for some reason never cared for in the twelve years I do with Linux.