Oh, I found it finally: There is a responsive shortcode for switching between different kind of displays. While the wp-table is not responsive, I used this shortcode to turn it off on tablet displays, where it does not display properly. That have caused my issue.
It should be ok now.
Hi, the table is here:
http://www.fors.cz/clenove-a-pozorovatele/seznam-clenu-a-pozorovatelu/
In addition to what I wrote above – I just checked Safari now and the filtering tool does not work there either. It appears only in Chrome.