Edit: I have updated in my post below that it still has problems and it isn't solved, as I previously mentioned here.
Thank you basj and sdRohan for your support. I followed the instructions to create a new Admin account, I reinstalled Fences under that account, I did a bunch of tests and everything was working. I logged back into my main account and things were still working, even after rebooting and reconnecting and disconnecting my external monitor.
It's weird that my system seems to be working with Fences now even though it wasn't before. The only configuration change I've done in the meantime is that I re-installed Nvidia Control Panel, because it was broken when Windows Update automatically installed it. That app is working now, and so is Fences.
I wonder if the other users above me, misc17 and glynnpg also have Nvidia display adapters that are misbehaving? I don't know how to tag them to let them know about this. They just have to manually download the Nvidia Control Panel app from Nvidia's website and install that to replace the version that came with Windows Update.
Anyways, thank you for your support.