When the switch is opened, and current stops flowing into it, it is a property of a coil, that it will try to maintain this current flow. To this end, it induces a voltage large enough to do so, and with this voltage it drives a current into charging the capacitor, spending the magnetic field in it during the process. When the coil is demagnetized, it is the capacitors turn to act, and it discharges through the coil, depleting the energy it had stored, and building the magnetic field back in the coil. Only a bit of the energy is lost due to the resistance in the coil.
Like a seesaw, this goes on and on untill eventually all energy is lost as heat.
This is a well known phenomenon, it is called the LC ressonant ringdown, and it was waveforms like this I was hoping to see.
I will have to do some measurements with the module in the bike to try to find out what I am missing.
If you feel like diving a bit deeper into the function of the good old points breaker system, here is the absolute best article I have found that covers the subject:
https://ttypes.org/conventional-ignition-systems/To be continued