Rein and whip handling took many forms in the pictorial record. Sometimes, reins entered a closed fist allowing for a bridged grasp that permits left and right steering through twisting the hand and wrist. The right hand of the driver depicted on the vase is a good example of a bridged grasp.
Other times, the driver uses a closed fist without bridging. Riders neck rein in this position, however neck reining does not work on driven horses because rein length and harness parts change the angles at which reins can contact the horse's neck. Also drivers lack seat aids involved in proper use of neck reining. If the driver can move the hand far enough, one can contact the horse's mouth on the opposite side. Example: moving the hand to the far left to tug on the right side of the horse's mouth. This method of steering would be very inefficient. Barring a written explanation, this method is likely artist interpretation and perhaps an attempt at depicting a method similar to the Victorian style.
By the early 17th century we have these two drivers displaying a Victorian/Modern Coachman's position, reins threaded through the fingers of the left hand. In this position, the driven can twist the hand and wrist to direct the team.