My classification of martial arts is even more simplistic but it meets my needs. I place each school, instructor and practitioners in one of two categories depending on which of the two is a better fit (it is not always a perfect fit). My interest is in training in the warrior traditions that have yet to be fully bastardized for commercialization, the ones that have yet to be neatly packaged for world-wide distribution or fully transformed into a sport or a form of entertainment or become a bunch of parlor tricks. I am OK with Warrior traditions that have sports and entertainment components as long as the Warrior tradition its self is still in tact and still the primary focus of training.
1. Warrior & Warrior disciplines
2. Other Stuff
I don't divide the Martial Arts into grappling vs striking etc because my general rule of thumb (not always the case) the less bastardized Warrior disciplines will be more well rounded and by necessity cover all of the ranges and a wide range of weapons, they may also include a competitive component and an entertainment component but neither would be the focus, while the more bastardized they become the more neatly packaged and narrow the scope. You can have the same martial art, for example Jujutsu and you could be looking at a version still in tune with their warrior past to include training in armor, using weapons and blending striking and grappling methods or a version in which winning sports competitions has become the only focus. You could be learning a Karate that has you shirtless, blending striking and grappling methods and wide array of weapons and conditioning or one where everyone has on a uniform covered in patches with various colored belts and emphasizing techniques to win a point tournament. So I don't lump all of any kind of martial art into one category but on a case by case bases as I encounter and examine it I will mentally classify it as true to the warrior ways or place it in the other stuff category.