Unfortunately, Dell does not test/validate this monitor model on Apple hardware, ports, and software. Because of this, Dell does not have many troubleshooting steps to offer. You could post on the Apple Discussions Forum and see what they say.

Test the Mac Mini HDMI out to BOTH monitors DVI in. If the message appears on both monitors, the monitors cannot be at fault. It has to be the cabling or how the Apple operating system is running the HDMI out.

I tried your test and the U2412M does not show the message, only the P2213 does.

What is worse for the P2213? Should I let it continue to power off and on when in sleep mode, or should I not allow it to sleep and enable a blank screen saver - thus having the P2213 in an always on state?

You could do the blank screen saver or get it exchanged and see if the replacement P2213 does the same thing.

Thanks. Just so I'm clear, you're saying that it would be less ware and tear on the monitor if I choose to use the blank screensaver and thus have the monitor on all the time as opposed to letting it turn itself on and off continuously when it's in power save mode?

Correct. I use a blank screen saver.