At the respective address in the browser you can see that the language filter is not taken into account in the search module.
Do I overlook an additional option to be set in or for the search module?
1/ Post a bug report at dev.tiki.org and wait for someone to fix it.
2/ Again I didn’t test but I understand from your post that the "Search page" do the job while the "Search module" don’t.
If this is correct you should be able to create your own module (custom module) using the "Search page" template code.
3/ Or... you could hire a consultant at : https://tiki.org/consultants to have it fixed for you.
4/ Use a customSearch module ?