Yes, you are correct. The cylinders need to stay upright. Which is why the small bottles are nice (for me). I transport them in the plastic tote that came with the kit. It keeps them secure from rolling around or banging into each other and I am able to keep them upright.
Obviously anytime you transport compressed flammible gas you need to be as careful as possible as there is always risk - much the same as when you transport 20 lb bottles of propane for the gas grill.
I always try to drive directly to and from the gas distributer with no side stops to minimize the time in the car.