Different voltage readings when using different ranges on a multimeter
There's going to be an AC component to almost any battery charging system. It's DC, but DC with some ripple on top of it. How much ripple depends on the charger. So that's why the readings jump around when the charger is active; the voltage is actually jumping around! You'd need an oscilloscope to see exactly what the waveform looks like. The DC reading on the meter is going to filter the ripple down, but exactly how it filters it, and to what degree, is going to vary from meter to meter and from range to range. The differences in this filtering are what's causing your different readings.

Unfortunately, I'll have to leave the details of how that filtering takes place in the meter to someone else...