At the beginning transistors fully ON (saturated) thus transistor collector voltage is then pulled down to near ground. And this means the primary winding will see full voltage across primary inductance (L1). Current will ramps upward, according to the ΔI = V/L*ΔtLet me summarize my understanding for you to see if I really get it.
The transistor start switching off(or start coming out of saturation) immediately when current flow in primary winding reaches the 'critical point' (that is to say that the current flow is no longer increasing exponentially at "I=V/L×t'' or in other words, no longer ramping up).
So the inductance of the primary winding decreases, and current flow in the primary winding will no longer be opposed,(as the primary winding now act as a normal conducting wires, current is limited only by "V/R").
So this in turn increases the current at the collector, which according to your first explanation will increase the collector-emitter voltage drop and force the transistor out of saturation and the transistor start switching off, then the primary winding will now induce a negative feedback voltage on the base winding which will reverse bias the CE junction and switch off the transistor.
And the current will continues to rise in the primary winding until Ic < β*Ib. Now the transistor will start to leave the saturation region and collector voltage will start to rapidly rise. This reduces the voltage across the primary inductance. It also reduces voltage developed across Auxiliary winding and reducing the base current for the BJT's, which now has to lower its collector current even more.
So, if the base current was reduced the transistor will increase the impedance from collector to emitter, attempting to lower the collector current in response to the lower base current. Thus the primary winding voltage will reduce further and we will see a further reduction in the Auxiliary winding voltage...->dropping transistor base current even more. This quickly turns into a case where transistors turns-OFF entirely. Do you see the Positive feedback mechanism here?