so, update:
I have read this interesting post:
Force CSS changes to “go live” immediately
What I did, is to add ‘?v123’ at the file functions.php, line 138.
The new line looks like that:
wp_register_style('main-css', get_stylesheet_directory_uri() . '/style.css?v=123', array(), NULL, 'screen');
now it works. It’s apparently a problem with my hosting company, that is caching the files…