Here is a schematic of the "15kV generator" kit with more details.
I think it's better mostly a saturated-core oscillator, as I understand how it operates:
Initially, the transistor gets base-current through the feedback winding and turns on.
Flux in the core increases and the feedback winding generates (in-phase) EMF to assist and add base-current.
Current ramps up until the flux no longer can build up- either the core saturates OR the transistor plus winding resistance plus battery reach their maximum current. The flux density is no longer increasing.
Then the feedback winding generates no EMF, and the transistor turns off slightly.
Any drop in flux density causes the feedback winding to generate opposition (counter EMF) to base-current which turns off the transistor, until the flux falls to nothing and the cycle repeats.
I did not measure the core's saturation point yet on these cores, I did not see an air gap.
I find these kits have crappy counterfeit transistors which tend to run pretty hot.