When you have a look at the first two plots, which I recall here:



then you can see, that the first plot shows the maximum of signal, which is a measure of punch, at a negative voltage, while the second one shows the maximum at a positive voltage.
This behaviour indicates, that your signal is the superposition of more than only one oscillation, the actual signal and some interfering resonances! Such a signal will not allow you to determine the punch by the help of rise time measurement. Also, the measurement of maximum voltage by the help of peak detector will not help either, unless you take both polarities into consideration. But then you still have to determine which peak is the right one, the positive or negative.

I would furtherly experiment with the drum pad to piezo interface, in order to get a stable signal of only one polarity, which is to be expected from a hit drum pad. Why not trying these super small and cheap electret micros mounted at the back of small enclosure, where at the top the drum "coat" is mounted? Your midi drum pad could look like a miniaturized tom, which is rather closed, and where at the top the coat is mounted and at the back the micro, sensing the pressure inside of tom. When you hit the drum pad then, you should always get a signal of same polarity, heavily easing the further signal calculations.

Kai