1) "never connect a low impedance voltage source to the inputs"
It means don't connect a PSU + and - to the input of the CR. Low impedance means low resistance in DC current, meaning that you would pass a lot of current at a high voltage difference (differential) between the inputs. Which would far exceed the 33mV max that is expected, and that would grossly overload the opamps in the CR.
2) "A maximum differential of 33mV is allowed before overloading starts to occur (vBat > 3.4V)." ?
The input current is measured as a voltage differential (created with the internal shunts) that is amplified 100x times.
IOW when you pass in 1AMP of current through the internal shunt of the CR, that shunt creates a 10mV burden voltage across that shunt (Ohm's law: current passed through resistor = voltage diff across the resistor), which is then amplified 100x times to result in 1000mV at the output of the CR.
Consequently, if you pass 3.3Amps through the inputs, that would create a 33mV differential across that shunt, resulting in 3.3V at the output. Anything over that, and the opamps are overloaded. If you had an ideal PSU that is set to 1V, and you connect PSU+ to CR_input+ and PSU- to CR_input-, then that would result in 1000mV across that shunt, overloading/saturating the second opamp.
So the bottom line is that this is a high side current meter, and you should always connect the CR_input+ to the PSU+ first, then have the DUT connected to the CRinput_-. So always have a DUT in there, because that's how it's supposed to work. Then heed the warnings about ground loops.
This is another reminder to try to find some time and create some diagrams with DOs and DONTs that should help explain all this verbiage with more visuals.