I solved the problem, thought I would post here in case it can help anyone.
To elaborate, my MINI S12 behaved as follows:
If I do Windows Restart, there is no HDMI signal detected by the monitor.
If I do Windows Shut Down, then press the MiniS12 power button to restart it, there is no HDMI signal detected by the monitor.
If I do Windows Sleep, then press the MiniS12 power button, or press the keyboard space bar, the monitor wakes up OK.
If I pull the power plug from the MiniS12, then plug the power plug back in, then press the MiniS12 power button, the monitor wakes up OK.
I own two MINI S12 devices, which allowed me to discover the cause of the problem by the process of elimination. For reference I will call the two units that I own MiniS12A and MINIS12B.
MiniS12A is powered with Power Adapter A, connected to Monitor A with HDMI Cable A, and connected via USB to Keyboard A and Mouse A.
MiniS12B is powered with Power Adapter B, connected to Monitor B with HDMI Cable B and connected via USB to Keyboard B and Mouse B.
MiniS12A is the one having the display problems. MiniS12B works ok, all as expected.
By swapping the the various components, I was able to identify the cause of the problem being Keyboard A. This was completely unexpected. Now knowing this, I suspect the keyboard had developed a fault that caused it to draw too much power. I did observe that the keyboard backlight was always very bright and could not be turned down.
From reading the forums it appears there are many possible causes of this problem. My cause is probably a rarity. Lesson learned, don’t rule out any possibility.