Should I turn off external monitor when MacBook sleeps?

Solution 1:

With modern monitors that have a "soft touch" power button they don't completely remove power from the monitor and draw zero current like say a traditional light switch. Part of the power supply keeps running to run some digital circuitry (normally a small microcontroller) that determines when the monitor should be taken out of sleep / off mode because either the button was pressed or a video signal is detected.

That would be why the specifications list 0.5 W for both modes. In reality there might be a very small difference to keep a LED blinking but that would be something in the order of 0.01 W. To put it in perspective 0.5 W x 365 days x 24 hours = 4380 watt hours or 4.4 kWh so in either mode you're looking at under $1 a year and it won't affect the lifespan either way because the modes are so similar.

As for turning on the monitor first I'm not sure but would guess that's just to give time for the monitor to start up so that the attached computer can detect it's present. It could just be leftover in the instructions from an older PC and/or operating system combination that required it.

Solution 2:

Will this save energy, money and the life span of the monitor - or the opposite?

Using no current as compared to very little current will save energy and money. How much, will need to be measured by you using something like a Kill-A-Watt power meter.

As for the lifespan, what you save will be negligible and most likely moot. LCDs now have a MBTF of 100,000 hours or more. This equates to 11.5 years of continual use. Chances are you will get a different monitor because your needs would have changed (several times) by the time it fails.

As for your edit/question, what that is telling you is that sleep and power off is the same except that in power off, it doesn't respond to the "wake" signal sent from the computer when in powered off state.