I consider a hot dog to be either the sausage (if served alone) or a sandwich (if served with the roll). The sausage is placed (sandwiched) between the two halves of the roll.
Wikipedia agrees with me.
A hot dog is a cooked sausage, traditionally grilled or steamed and served in a sliced bun as a sandwich.
The Oxford English Dictionary cites these AMERICAN sources:
1901 National Mag. Nov. 168/2 He seated him upon a high stool and presented him with a ‘hot dog sandwich’.
1917 E. A. Guest Just Folks 68 Though twenty miles seem far to us, it's just a hop to them Who yearn for hot dog sandwiches at two o'clock A. M.
1922 National Druggist Nov. 475/1 If you want a hot sandwich of your own, we have the frankfurter, better known as the ‘hot dog’, which may be cooked on the premises.
A sandwich is a food item consisting of one or more types of food, such as vegetables, sliced cheese or meat, placed on or between slices of bread, or more generally any dish wherein two or more pieces of bread serve as a container or wrapper for some other food.