Change directory for Kate syntax highlighting files
I found a solution just by chance:
Settings -> Kate config -> Open/Save -> File types
If I click on "Download syntax-highlighting files", all files which are already installed and can be updated (a newer version is available) are marked here. Then a click on "Install" updates alle these files. After that the folder
~/.kde/share/apps/katepart/syntax
exists and includes all installed XML files. Now I can add additional files to that folder or can modify the existing files therein. The modifications will be applied after restarting Kate.
This solves my problem for the moment. But I don't understand why it works in this way! (Maybe someone can comment on that)
The other open task is how one can choose a different folder for the XML files.
For the latest version of Kate, placing syntax xml files in this directory works:
~/.local/share/katepart5/syntax
Thanks @Datalore