My opinion, the two systems are both auto-dry. More correct to differentiate them as thermostatic auto-dry and moisture-sensing auto-dry.
Thermostatic in my experience is not consistent when the dryer is installed in a non-climate-controlled space such as a garage or porch. Cold weather causes longer run-time and over-drying trying to reach the target temperature to run the timer out to Off, while the clothes are still drying, giving up moisture during that process. Very hot weather skews the other direction. Moisture-sensing is more consistent via directly reading the moisture level of the fabric without dependency on hitting the temperature point to run-out the timer.