The 'jumping' issue is actually kind of intended behavior, as the search position becomes 'fixed' to a certain spot on the page, while the results position remains absolute. The only way to compensate that is to change the search position every time a scrolling event occurs. However that can be very resource heavy operation, so a delay is added, to wait for the scrolling to finish, and then move the results bar below the search bar again – instead of moving it non-stop during the scrolling. It's basically to prevent lagging during scrolling effect.
I was not able to log-in to the back-end with the details above, I'm guessing those are SFTP details, but I don't want to log-in to that just yet, unless it is neccessary.
But I think I might know what the problem is. I see you mentioning tags, but actually those are product tags, which is not the same taxonomy. I'm guessing you selected this option on the index table panel: https://i.imgur.com/ggAspuB.png
However those are not the product tags, you will need to choose this option: https://i.imgur.com/kECBUG3.png
After saving the changes, make sure to click the 'Create new index' to re-fresh the index table.
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