Why can the loading resistor and capacitor let the unloading waveform become from negative to positive voltage?
A piezo-electric crystal generates an AC voltage charge under acceleration. There is no DC content.  With no restriction on motion under a sine acceleration yields an unloaded sine wave.  

A cantelever PZt is alike a diving board with end stops so it generates a sharper step voltage upon impact, shown by the loaded "jerk" or rate of change in acceleration and change in direction of acceleration and also abrupt change in direction by the rebound.

By superposition you can imagine a square wave added in the opposition direction after the peak acceleration declines. this indicates the end stops are on board sides of the cantilever causing a reaction force and acceleration to the PZT vibration.

Since velocity is the integral of a(t) a 90 deg lag would indicate the PZt was approaching the maximum velocity before impact but did not as the force was still acclerating before it hit.

this stiff end-stop allows a stiffer voltage and current to be used to charge the inductor to which the stored charge energy is being transferred into a stored rising current, which is then commutated when the poarility changes direction to result in a continuous DC charge curren to charge the battery.

The challenge of this design is to make the PZT robust enough to survive expected mechanical energy being harvested and converted into voltage, then switching into alternating  inductor current which is the integral of the applied voltage or a loaded sawtooth to a loaded square wave with a weaker sinusoid.

Why is the sine weaker?

Because we know the PZT capacitance produces current according to Ic=C dV/dt and the square step yields a high dV/dt than the slope of the sine wave.