Hmm. I rechecked this theme's details and was reminded that the Mittwoch theme's tiki-site_header.tpl file (not the custom code) contains one search form, the simpler one, conditional on "if $feature_search eq 'y'". The control for this is the "Search" checkbox on the Admin Features page.
The other search form, with the dropdown to narrow searches, is the one activated by the search bar checkbox on the Site Identity admin page and is displayed "if $feature_sitesearch eq 'y'". I don't know why it's displaying at your site because there's nothing in the Mittwoch templates that triggers it (the one at doc.tw.o is displayed by the site identity custom code there).
Sometimes theme changes aren't complete until you clear the template caches. Did you use the System Admin (Exterminator) (tiki-admin_system.php) to clear the caches after the theme change?
A second problem I notice with your screenshot is that the site title has the wrong styling. The h1 in the site header is using the styling intended for h1.pagetitle. I'm not sure why this would happen unless something related to the search form problem is interfering with CSS selectors or something. I have Tiki 1.9+ at a couple of site and am trying to replicate these problems but so far haven't been able to.