This is incorrect when it comes to the regulation (what Tesla thinks is irrelevant). Read the linked text. Specifically the definition of "Autonomous mode". It quite clearly says that both with or without a natural person or remote controller monitoring.
@ReflexFunds
"An autonomous test vehicle
does not include vehicles equipped with one or more systems that provide driver assistance and/or enhance safety benefits
but are not capable of, singularly or in combination,
performing the dynamic driving task on a sustained basis without the constant control or active monitoring of a natural person."
Nags means you are not an autonomous test vehicle and are not required to report anything.
No, that's a misrepresentation of what is stated. The distinction is made on designed capability, not on the actual mechanics of the driver controlling the car. Also, even then it is not the nag but the physical handling that is required that shut up the nag that plausibly could have been the defining marker as you correctly identified above when you claimed it's "Hands on wheel" that makes the distinction. The DMV made specific changes in November 2017 to exactly close that loophole by replacing the wording of "active physical control" with "supervising the autonomous technology’s performance of the dynamic driving task".