You can actually use a martial art in a "real" fight. In a fighting sport, you know that your opponent follows the same (arbitrary) rules as you do, so the fight is formed around them instead of, well, fighting.
I don't think there is a meaningful distinction. All forms of training and competition* in hand-to-hand combat involve "arbitrary rules", because it is a bad idea to kill or maim your opponent. When I practice the taiji form "White Snake Turns And Spits Venom", for example, it is obviously unacceptable to crush my training partner's larynx with a blade-hand strike, so I "pull" the blow. We are not competing for points or coloured belts, but our shifu will praise us be grudgingly less critical if we perform well in front of him, which is not so different. In modern wushu competition, taiji is judged purely on form for points (often to music), rather like gymnastics or figure-skating, something that many traditional shifu regard rather negatively. Many practitioners of taiji only ever perform the slow solo forms for health benefits, and as a spiritual and physical discipline. So, is taiji a martial-art, or a sport, or something else entirely? I would say that it is all of the above.
*Obviously a serious fight to the death or disablement of one or more of the participants is a form of competition, but that is not normally what we are talking about, and I have absolutely no desire to inflict injury on anyone. There is perhaps a matter of intent here. If a person trains in taiji, or krav-maga or boxing or whatever with the intent to win points and prizes in competition, then regardless of the fighting-style chosen, the sporting aspect is obviously paramount. Any contest that takes place in front of an audience nowadays is mainly sporting, and even the more hairy-chested styles like muay-thai or MMA have "arbitrary rules" about how the combatants are allowed to fight because having fighters killed in the ring is normally not intended or regarded as desirable. There is no doubt that this affects the judgement of which styles of fighting are more effective. The widespread perception nowadays that grappling is more effective than striking has much to do with the fact that many of the most effective strikes are banned and/or made impossible by protective equipment in competition, and grappling is safer in training. Suffice it to say that if I ever had to defend myself against an assailant trained in boxing, I would not underestimate my opponent because boxing was a "sport".
While you make excellent points I will continue to disagree, I know we have discussed this extensively before, but it is all intent, approach and mind set. Though some forms of "fighting sport" would not translate in the least to the real arts they mimic. I am lead to understand that a kendo practitioner would have an extremely difficult time with Iaido or Kenjutsu (and an Iaido pracitioner would have issues with Kenjutsu, though not as much as the Kendoka) as kendo has drifted so much in technique and practice from the actual martial techniques of the Japanese sword as to barely be connected. Similarly Olympic Fencing has about as little relationship with proper training with the rapier, small sword, saber, or any other European blade as to have almost no appreciable connection beyond a historical one.
You're right rules in point sparring/competition do have their place. We don't want to kill each other or severely wound each other but the attitude that point sparring competition is a focus of martial skill as opposed to an amusing diversion to promote camaraderie and test skill amongst fellow students and against other schools is unhealthy in my opinion. Many schools and systems train and focus FOR their sparring competitions directly. A martial discipline for combat or defensive use should teach and maintain that mind set most of all. For combat the mental development of the student is key, The Marine Corps Martial Arts Program (MCMAP) uses the slogan "One Mind, Any Weapon" as it's motto and one of it's fundamental tenants. From the ground up it is designed to make the human body a weapon, like all martial arts that proceeded it, and then ensure that the Marine or Corpsman can fight effectively with any weapon at hand or in their possession. Warriors have been trained this way for thousands of years and it is no less useful now.
For the civilian martial artist, a serious mind set and strong mental focus are important if you truly intend to rely for your life in your training. If your training is just a form of exercise to you, the question of art or sport has already been answered from the very beginning.
"Since karate is a martial art, you must practice with the utmost seriousness from the very beginning."
-Funakoshi Gichin Sensei
"The heart of our karate is real fighting. There can be no proof without real fighting. Without proof there is no trust. Without trust there is no respect. This is a definition in the world of Martial Arts."
- Masutatsu Oyama Sensei (commonly known as Mas Oyama)
"Although it is important to study and train for skill in techniques, for the man who wishes to truly accomplish the way of budo, it is important to make his whole life in training and therefore not aiming for skill and strength alone, but also for spiritual attainment."
- Masutatsu Oyama Sensei
For the record, while I have a very deep respect for Oyama Sensei, his accomplishments and his work, it should be noted that he regularly fought bulls with his bare hands. I'm a proponent of serious study of the martial arts, but Oyama Sensei is a bit extreme even for me.
On a slightly related note, last time I got x-rays done my radiologist noted that aside from the three fractures, the bones in and around my striking surfaces and their supports are quite dense compared to a normal human being. Not near what a truly long term master (or student for that matter) would show, but I found that extremely gratifying.
For those interested in the science behind the human body and martial arts, I recommend the show Fight Science.
I will leave with some unrelated words from the Shodai Soke of Goju-Ryu karate, Miyagi Chojun Sensei.
Oh, and Akima if you ever spar a boxer, either get inside his range and use knees/take downs/ and other close strikes or take the long range route with kicks to the lower body. A boxer's powerful punches are his strength but also his limitation, he will not be used to fighting with or defending his legs. Boxers also fight in spurts, this can be used to your advantage if you can weather the storm of his initial assault.