I would like to be able to set "Extend my Windows desktop onto this monitor" via code. A PowerShell script would be ideal. WMI seems the way forward but I have zero knowledge in WMI.
Windows 7, 8 and 10 are supposed to come with a small program that does exactly this: displayswitch.exe. This page lists the following parameters:
displayswitch.exe /internal Disconnect projector (same as "Show only on 1" from the Display Properties dialog)
displayswitch.exe /clone Duplicate screen
displayswitch.exe /extend Extend screen
displayswitch.exe /external Projector only (disconnect local) (same as "Show only on 2" from the Display Properties dialog)
For a one-click solution to the problem posed, simply create a *.bat-file containing the single line
call displayswitch.exe /extend
and save it to your desktop.
[I tested this on Windows 8.1, and it has been confirmed to work on Windows 10.]