Can I boot my mac without pressing the power key?

Back in the day you could start a Macintosh using the power key on the keyboard (like the Apple Extended 2 has one in the top right).

These days we have many (more) input options like: Power, USB, Ethernet, HDMI, Thunderbolt, Audio in/out, SD card reader, Wi-Fi, Bluetooth.

I have a MacMini (2011) which is under the desk. Is it possible to boot a mac (or my Mini) after a shutdown WITHOUT pressing the power button?

I tried the 'Start up automatically after a power failure' but after a cold shutdown, a power cord disconnect/connect did not boot my computer.

shutdown -u

I tried the 'Start up automatically after a power failure' but after a cold shutdown, a power cord disconnect/connect did not boot my computer.

The automatic power on only works if the power was cut. You can induce this behaviour using the command shutdown -u now. According to man shutdown the -u flag does the following:

-u: The system is halted up until the point of removing system power, but waits before removing power for 5 minutes so that an external UPS (uninterruptible power supply) can forcibly remove power. This simulates a dirty shutdown to permit a later automatic power on. OS X uses this mode automatically with supported UPSs in emergency shutdowns.