Intuitively Sal's comment about the wider current pulse meaning the capacitors are more discharged makes sense to me.  When I look at the details more closely, very little with the Onan trace makes sense.  The Onan voltage has a pretty long duration flat top.  If the caps were discharged I'd expect the current to peak synchronized with the voltage hitting its peak.  But it doesn't, and instead has a nice gradual rise.  If the converter was short on incoming power I'd expect the current to stay up there until the flat-topped voltage drops, but it doesn't.  The converter's regulation looks "stupid" to me, just like the generator's output.  I don't know the details of how the PD regulator works.  Is it as simple as mutual confusion between the generator and converter regulators?
Compare that to the sine wave (Honda) traces where everything appears as it should.  In comparing the current pulses the only thing I'm confident in saying is that because of the squaring, the shorter taller Honda pulse likely has a higher RMS value than the longer wider Onan pulse.  I was thinking I could re-run this with the new scope that gives an RMS readout, but that would just give VA.  I don't think there's an easy route to measure watts.  (Don't have time now to do it anyway).
For these shots I was running the scope uncalibrated to make the picture pretty.  The unknown scale was the same for the Onan and Honda shots.  I also saw in my notes that with the PD heavily loaded, adding a resistive load with the Honda drives its output down.  With the Onan, adding a resistive load results in the PDs output going up, but even with the water heater running, the output with the Honda is higher.