I agree with all 3 points on principle.
However, I would say that Martial Arts encompasses all 3 categories at the same time. The root of it all comes from the literal translation of jujitsu which is "the art of adaptability"

1. Martial system for self defense. Yes. Almost every combat sport is somewhat useful in some ways, even horseback archery (although it is no longer practical)
2. Martial sport for fitness and competition. Yes, combat is the ultimate competition, however one must realize there are no rules in self-defense, nor is there a point system. Winning equals coming home that day to feed the cats. There are no trophies for living to fight another day. You simply live to fight another day.
3. Personal development. No matter what style or school you choose, there is no "wrong answer" in martial arts. If you are challenged beyond your former capabilities, you have 'won'.

My advice: choose a school that matches your personality. Are you a brawler? Boxing and MMA and Karate call you. Do you desire peace and personal development? Qi Jong or Tai Chi.

If you want something that samples every form of martial art, to include Kung Fu, Jujitsu, Qi Jong, knife, gun, sword, stick, healing massage, acupuncture, herbal medicine, and fencing, check out Dan Zan Ryu.