You are right that the Hammond and some other organs generate discrete notes that can have their specific frequency modulated by a low frequency - which effectively is like an oscillator that has its oscillating frequency varied, and equivalent to say a guitar string that has its resonating length varied by a finger.
To apply a frequency modulation to a signal passing through an amp, the common electronic techniques are that used by Magnatone, and by Wurlitzer. The modulation is signal frequency sensitive, in that the level of modulation is maximum with signal frequencies around 1kHz and drops off for lower and higher signal frequencies (1kHz is typical centre frequency for guitar use, but Wurlitzer used a lower centre frequency for its organ notes).
The ear hears the same modulation, but a spectrum analyser shows they are different - for example a 1kHz signal modulated by 10Hz will show as a 1kHz signal with discrete 990 and 1010 sideband signals from the Maggie, whereas the Hammond would likely show a flat topped peak from 990 to 1010Hz with steep sides (although I haven't done that spectrum measurement). So I agree that 'true' pitch shifting was a little marketing slight of hand.