You've pretty much described it. The two pole armature gets current induced in it by rotating inside the stator and that is made to be DC by the diode or more likely, diodes. That then acts like the permanent magnet rotating inside a stationary magnetic field. There doesn't need to be a smoothing cap inside to make it pure DC. I'm not going to claim any knowledge of the details but the scheme certainly seems plausible and with care in designing the magnetic components seems like it should work quite well.
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