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The storage of the energy occurs during the ON time of the MOSFET cycle because current flows through the drain inductor during the ON time (Current in an inductor is analogous to voltage across a capacitor). During the OFF time, the energy stored during the ON time causes the drain inductance and capacitance to initiate ringing. The parasitic diode from drain to source causes the negative excursions of the ringing to be clipped off. This clipping of the ringing causes the stored energy to be dissipated.
The same thing does not happen with ordinary transistors, JFETS and vacuum tubes. What I am describing is called the “turn-off behavior” of power MOSFETS. It is described in detail with scope photos on page 5-41 of the Siliconix Mospower Applications Handbook, ISBN 0-930519-00-0. I have seen clipped, positive-going, pulses on the drains of Power MOSFETs used for part 15 AM transmitters that match the scope photos published in this book.
A circuit to isolate the MOSFET from the tank during the OFF time would be just what is needed to prevent dissipation in the parasitic diode, but I don’t know how to make such a circuit.