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. 

6th-8th Century BC Vases, Kunsthistorisches Museum, Wien 

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. 

Pg 46 - Magnus Olaus - Historia de gentibus septentrionalibus, earumque diversis statibus, conditionibus, moribus, ritibus ... necnon universis pene animalibus in Septentrione degentibus, eorumque natura 
KBR Ms. 9987-91 Hamartigenia & Psychomachia North France 890 114v 
Weiditz, Christoph. Authentic Everyday Dress of the Renaissance. All 154 Plates from the Trachtenbuch, 1529, 1531/1532. Reprint 1994. Dover Publications, New York. ISBN: 0 486 27975 8 
Hs 22474 - Christoph Weiditz - Trachtenbuch - [1530/1540] GNM LINK
Prince Maurice and Federick Henry at the Valkenburg Horse Fai Adriaen Pietersz van de Venne oil on panel 1618 

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. 

Folger Shakespeare Library - ART Vol. c91, no.7a - cd title "Royal, and Military and And and Court and Costumes and Of and The and Time and Of and James 1"