Stagecoaches in the American West, were a totally different design than Park Drags or Road Coaches found in the UK and the East Coast States.
Then there are sub-stagecoach catagories, of various body styles, used in the West and East before railroads had much track to distribute goods and passengers. The most commonly known is what was seen in movies, made by the Abbott and Downing or Concord Carriage Companies in the East.
Their design used leather “thoroghbraces (sp?)” as slings to support the body on a running gear. Somewhat similar but not exactly the same, to the Gold State Coach body hung on leather straps. The American design had about 12 leather strap layers under the body, with the driver’s seat fastened to the coach body, which had it moving constantly as the stagecoach traveled along. The boot in the rear of body held luggage, mail, goods, which probably helped balance the body on the straps. The driver had a loose rein connection to horses, due to the rocking motion during travel. He/she could not maintain a steady mouth contact over the rough ground. Roads could be usable or disappear seasonally, nothing like maintained, sometimes surfaced roads in the UK that allowed fast stagecoach travel. These were very tough vehicles, heavily built, braced, to survive the conditions they were driven over.
European design seemed to often have the driver on a “perch” above the wheel and axle running gear, unconnected to the slung body where passengers rode. One of the articles mentions a King having the drivers seat removed from the Gold State Coach in the past.
I would call the pictured red and black coach as likely a private vehicle. It may not have carried people inside! Servants don’t count. All the friends would be riding on top to the excursion or picnic outing. Spare parts, servants would be inside, food and drink in compartments, coolers, in the rear behind a fold-down tailgate/table to serve off of. Once parked at the destination, inside the coach body was also probably the loo, since porta-potties had not been invented yet! We have seen a number of private coaches fitted out this way, with a potty.