I have to disagree. The 3 overlap sometimes but they're not the same. Examples:
1. If I'm wining a Kata competition, I am practicing a martial art but not fighting or self-defence.
2. If I'm pepper-spraying a mugger, I am practicing self-defense but not fighting or a martial art.
3. If two bums are fighting on the street, they are fighting but not practicing martial arts or self-defense.
There is a clear distinction in the above scenarios.
kata competition isn't martial arts imho, just like XMA and tricking aren't martial arts. even by simple dictionary or wikipedia definitions they are not martial arts. however the entire point of kata is as a training method to help teach people to fight, so i could see some argument for why kata competition is a martial art competition...but to me if your end goal is winning kata competitions, you are not a martial artist or training a martial art. if the point of the kata competitions is to further push and refine fighting techniques, whether you ever use those techniques in a fight of any sort is irrelevant, then kata competition is a practice of martial arts.
i didn't say that all fighting was always martial arts. so you're right, two bums fighting over a can of spam isn't martial arts. no one said it was.
however, if you trained use pepper spray against an attacker that is training a martial art. that martial art may not have a formal name like karate of muay thai, if you regularly practice and train with it, even once a month that meets the definition of training to fight.
and to clarify a few things. if bill wallace only trains kata these days for health, he is still a martial artist, because he spend years fighting, and training to fight.
if a dude spent 20+ years training for self defense and never fought in any form outside of maybe sparring, but now trains just for health i'd still call him a martial artist.
i also don't think effectiveness of a martial art, or martial artist or perceived effectiveness makes a difference. so if you train aikido or JKD for self defense, you're practicing a martial art and are a martial artist.
so heres how i see it, martial arts=fighting self defense=fighting martial=/=self defense fighting=/=martial arts, and fighting=/=self defense
does that help clear things up and make sense? i tried to cover the nuances of my opinions and thought process on the subject fairly succinctly, but i'm sure theres a 100 'what if' scenarios i didn't cover, but you should be able to extrapolate my answer to them from what i've given.