So how does flipping the meter do anything if the output signal is AC with no offset?
I guess that implies the LCR meter is only sampling on the positive half of the sine wave, for cost/simplicity reasons.
Could potentially have a more advanced LCR/component tester that samples both halves and tells you what it sees in either polarity.
I would not expect them to sample only one polarity half the wave: it does not same much on the cost side and makes it sensitive to DC offsets. So it is more like save costs by sampling the whole wave and this way don't worry about DC offsets and save there.
It depends on the meter if they have an DC offset or not. Also the amplitude of the test signal can vary. So some meters can be more suiteable for in circuit test than others.
Changing the leads can still effect the effect of parasitic capacitance. Chances are one of the leads is more sensitive to parasitic capacitance than the other. This can especially make a difference when measuring in circuit or with small capacitors (e.g. < 1 nF).