Without an equivalent circuit with all relevant inductance, it is hard to analyze because appears to be a primary and secondary resonance.
Your secondary 5.6 ohm/5= 1.1 and 470//470pf ~ 1nF with unknown L.
But with 10 nF added the waveform has a Q ~ 10 with 10 cycles
when 1st peak decays to 6% which is the rule of how to measure Q on a 2nd order underdamped waveform.
I checked my RLC impedance graph which told me to damp the secondary instead of 1.1 Ohm+ 1nF you need 22 Ohm + 10 nF but this does not filter out the primary resonance. with unknown LC parameters.
If you can create a more realistic RLC model, I find your optimum damping factor. This simulation has a test switch to compare plots with 2 values of R.
https://tinyurl.com/ym247o8u
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