"local in nature" is considered a quick drive to a hotel from airport  

commutes to/from work are on you because it is from your resident to your domicile place of work.   IOWs, you are domiciled (based) at X, so your drive from home to X is on you.   If you drive to Y instead of X for your employer, it is no longer "local in nature", and is not considered part of your rest.

If you are co-domiciled (ie, assigned to two or more places), then going to/from home to any of those places is considered "local in nature" provided it is reasonable.

"local in nature" has no fixed distance requirement, and is not a universally applied radius from your domicile base elsewhere.   It is duty to ground transport yourself (drive, ride, bus, taxi, walk, etc) from your domicile to another, or from your home to a place other than your domicile.

You are on duty!  And, you cannot count that as part of your regulatory rest requirement (which includes counting it as a "day off")