Right - the unit appeared to be on which is why I didn't originally look for a power switch. Pressing the little button on the side will turn the led screen on/off (after initial plugin to the wall and boot up, the screen is on then counts down and goes to sleep). Obviously car charging goes on/off with the screen. The vertical green line is always on, and changes length depending on whether or not it's plugged into the car. Not how I'd expect the power button to work, I'd expect all or nothing, but now that I know I'll just leave it on and not worry about it. My wife has a TV in her office that despite me making the occasional comment is on 24/7/365, lots of other electronic stuff always on, and being a big (not huge) old and probably not well insulated house the central HVAC systems get a work out. In the grand scheme of things I can't image an led screen and whatever monitoring it's doing to show up in my electric bill.
Here is the important part now that I know how to turn it on. I plug it into my car, it charges my car at ~32 amps. Later, the battery in my car is full.