Hi Mark,
Thanks for the detailed description, especially the screenshot of your actual keyboard!
As you can see, the displayed diagram in Ukelele has an extra key between the left Shift key and the key where the Z is in an English-language layout. This is usually called an ISO-type layout.
But your hardware keyboard is not ISO-type, and the mismatch is what has caused your problem.
The fix:
In Ukelele, choose (from the View menu) View > Keyboard Type… and then on the right side in that window, change the drop-down menu to read ANSI.
(In the list on the left side, you can almost pick any type, but as an example, you can pick Powerbook USB Internal.)
That changes the display, including the key top left on the keyboard, just under the ESC key.
Now edit that key to match your intended use for it.