It appears that after a number of days of getting a 404 response code on the notification XML file that Windows 8.1 will finally give up and go back to just the tile image and text. See notes on the original post. I first pinned the site on Wednesday around 17:45 then turned off the notification XML file so that it returned a 404 response code. As of Sunday it was still showing the missing notification on the desktop. Monday at 8:45 it had reverted to the tile image.
Note: my Windows 8.1 is a Virtual Machine and does not run all the time. It's possible that Windows might give up on the missing notification sooner if it ran all the time.
Windows appears to look for the notification XML files when the user first logs in and whenever coming out of the screen saver.