Well, there are many opportunities for overdrive in even the standard Hammond works. Its built-in valve preamp picks up vibrations from the motor driving the tone wheel generators, and the spring reverb unit does the same -- both are certainly microphonic in the acoustic sense.
Jon Lord used to do unique things with the Hammond's output. Latterly (maybe 72-ish onwards) I think he had a custom preamp which included a ring modulator, and an output split between Leslies and "straight" amplification, with a pan control. The Leslie design is of course the source of a whole slew of astonishingly benevolent colouration in it's own right, with more motors and valve power amps in the same box, and the standard treble horn driver is often replaced with an Electovoice unit which stands up better to continued Lordism.
When playing lead, Lord was as you say exceedingly sharp, but he tended to choose the right notes to go with this. Equally, his swirly chords could thicken the background superbly. I reckon his starting point was almost Steve Winwood's approach (SDG: Gimme Some Lovin; intro), and he just developed it from that.