S.W.
I have trained in TaeKwonDo and Karate for a total of 10 years. I have a black belt in karate. My daughter started at age 5 in Karate and is now a Brown Belt at age 10. I enjoy the TaeKwonDo/Karate style of Martial Arts because of the emphasis on kicks and the physicality of it.
I have taught a kids' class for two years and I think Martial Arts are a tremendous benefit for almost all kids. Your son sounds like a good candidate for this sport. The kids that tend to do well are typically independent and somewhat competitive, and don't always like team sports. Stubborness is OK :-) My daughter fits this description (she loves swimming,too), and her stubborness will carry her through her black belt test process, which is grueling. The confidence and strength she has gained is noticeable.
I think visiting several schools and observing a class or two would be the best way to understand the differences and get a feel for the type of instruction offered. A school should not have a problem with you and your son visiting, asking questions, and taking time to decide. A too-pushy sales pitch would have me moving on to another choice. An instructor for a kids' class should be strict but not overly intimidating and also allow for a little fun during or at the end of a class. The teacher should know and use each student's name. Respect should be taught by showing the students respect, also.
Be sure to get the financial information, and be sure you know what is included. Some school charge a monthly fee for as many classes as you want to attend, some charge extra for each belt promotion, tournaments are usually optional and come with an added fee. Sparring equipment might be included in the overall fee or may be extra. Typically sparring equipment must be purchased through the school or only from approved vendors. We started sparring when we reached Green belt.
Good luck in your search!