Hi,
After checking the situation on your website and doing some debugging, I conclude that you have some mechanism on your website/server which redirects / modify the URLs on the fly (like a htaccess can do).
Because of that, the URLs to refresh the filter module areas don't retrieve the new HTML to refresh the area and thus it falls back on the previous HTML.
That's why, when you access the page
jouwnaambord.nl/naambordjes
and use a checkbox of the Materiaal filter, you don't get the selected element with the cross icon, and similarily, when you access the page
jouwnaambord.nl/naambordjes?limitstart=8...iaal_13=246&filter=1
and use the cross icon, the filter module doesn't display the checkboxes with all the choises, and revert to the previous display.
So I would recommend fixing the issue with the thing on your website/server which does that. But I don't know what it is.
Alternatively, an easy solution would be to turn off the "deletable filter" setting of your filter. That way, you would always get the checkboxes and even without the filter module area refreshing, it would still work as if it was normal.