Hi,
Thank you for the details, it helps a lot.
Based on the screeshots, you were editing the wrong options. The options on the screenshot are for the taxonomy term filters – not related to the ordering of the results whatsoever.
Ordering by taxonomy is not really possible, because taxonomy terms can be assigned to multiple post type objects.
For the live search results you can use the results grouping feature – where you can choose grouping by categories, and the chosen category order order is going to be used in the output as well. That should give you exactly what you need, but that does not apply to the results page.
The output of the results page is not affected by the plugin, it can’t. If you see any issues there, you should check your theme configuration, that may reveal the problem.