The "Update Search" button is dangerous in that it will modify a saved search without warning of the save. Since there is easy access to edit of searches via the "Edit Searches" button, my opinion is that the button should be removed. If you do not agree with that then more control over the update needs to happen such as an "Are you sure?" feature or redirecting to the "Edit Searches" page with the edit item function open with the new value to allow the user to "Save" the search on that page. Under no circumstance should the "Update Search" button change the search without confirmation. I've already had to correct searches more than once because of this button.