Okay, I have a foundation to work with here--two switches controlled by my signal generator. One charges the cap; the other fires it--basic CDI. One side of the coil and cap remain grounded at all times.
For the two scopeshots, the first one is pretty self explanatory: Yellow trace is charge, pink is fire.
In the second scopeshot, yellow is the ring-back on the exciter coil; pink is the signal on the collector coil. Notice there is some power transfer going on here, the output LED is lit. Exciter coil is on the bottom; collector coil is on top.
The capacitor I'm using is probably way too big, 100uF. I'll try to tune it better, but what's interesting is the ringing on the exciter coil--it's almost as if it doesn't care about the capacitor and instead rings based on its own internal SRF. That ringing changes drastically if I lift up the collector coil. Back EMF shoots way up even though I'm only charging the capacitor with five volts. The frequency and Q-factor of the ring also changes, so the collector coil is definitely communicating with the exciter coil--they don't work independent of each other. The fields emitted by both coils are entangled. Vortex rings mixing maybe...?