As for how the headstock-mounted Min-etune/Tronical system detects the strings' pitches, the tuning machines obviously must incorporate some sensor technology which senses each string's vibration and transduces it into an electrical signal which is sent to the control unit and processed for pitch detection and compared with reference frequencies, based upon which the tuning machines are driven to adjust the string tensioning.
Btw, I believe Gibson's original robot-tuning system used the guitar's pickup for pitch detection, so the shift to a headstock system no doubt required a bit of development in order for the motorized tuning machines to be able to reliably sense each string's pitch across the nut (and string trees). Although I suppose it's also possible that the system senses the headstock vibration and isolates each string's pitch through DSP filtering rather than employing sensors in each tuning machine.
The Peavey/Antares auto-tuning is "virtual" using DSP rather than actually phsically varying the string tension, similar to what Roland's V-Guitar systems have already been implementing since the 1990s. Just do a search on "VG-8" and "Joni Mitchell".