Briefly: the distance from the underside of the outboard's mounting clamps (the point which actually sits on top of the transom) to the horizontal cavitation plate located just above the propeller. this dimension will be about 16 or 17 inches on a "15 inch" motor, and will be about 21 or 22 inches on a "20 inch motor." So, yes, it should work.
The definition of long shaft vs short shaft o/b engines has changed somewhat over the years. It used to be that o/bs designed for 15" transoms were "short shaft'" and the "long shaft" engines were 20". Now, with the popularity of deep Vee hulls, 25" transoms are seen. So, "long shaft" may refer to 25" lower units, particularly for very large hp engines.
For more info see: http://www.duckworksmagazine.com/05/columns/max/4/index.cfm