It has always been my understanding, that when a driver is "away from the HOME terminal", his assigned tractor can be used for personal conveyance, as long as there was not a trailer attached.
That was how it was treated back in the 70's and 80's, and how my last "company" job and my current "lease" position treat(ed) the situation.
Bobtail the tractor as far as you want "off duty not driving", but if there is a trailer on the unit, loaded or empty, every mile had to be logged.
Did you ask the TMC log department what their preference is Terry??