Yes, there were long and short C/30s. It's wrong to call them short beds and long beds since beds didn't come on them. In the truck world you go by wheelbase , or cab to axle measurement for the body. C/30s were 133" and 157" WB, or 60" and 84" CA. Trucks are cab & chassis vehicles with the body being whatever is mounted on the chassis for truck's purpose. Pickups were an afterthought as people started adapting cars to do more than haul people. There were C/30 pickups and, although that bed was longer than any other pickup bed offered, it was only offered on the short C/30 chassis. Most all C/30 tow trucks were built on the short chassis and they look even shorter since a tow body is bobbed as close to the rear spring hangers as possible, for obvious reasons. The same is true for dump bodies, since the truck can handle only so much weight and they are bobbed to keep the pivot close to the rear axle
Here's my old '71 C/30 133" wrecker body
And my old '70 C/30 157" w/11' utility body