Generally....What is given is the area available for heat transfer. It is usually based on the outside diameter of the tubes. You can usually take the outide diameter of the tubes times the length of the tubes times the number of tubes and get the same answer as the "surface area of the exchanger". That being said if the heat exchanger is a U tube bundle is is much harder to back calculate due to the nature of the design.
I have never seen the tube area and the shell area added together to be called the "surface area of the exchanger". The only time that would matter is if you were having a coating applied to the tubes and shell and they were charging you by the square inch.
Be careful that to check to see if it is indicating total surface area, or effective surface area. Some surface area is blocked by support plates, baffles and the like.
Surface area usually is limited to the tubes only, and if a U tube, only the straight tube portion. The bends are not considered, nor is the surface area of the tubesheet nor shell surfaces. If these help your heat transfer, then they are just lagniappe.
