If I could, I would like to summarize some points that have been made and get your opinions on there validity to the original question of being fit to be called a martial artist.
1. Fitness will improve your performance in martial arts.
2. Fitness will improve your life.
3. Training will make you fitter.
4. To be considered a martial artist you need to pass some standard of fitness assessment.
5. You can be a martial artist even on your deathbed.
I can understand some of your criticisms of extremely out of shape and poor performing instructors. I have never encountered this within my art. I have certainly encountered instructors of varying fitness levels, but this had little to do with their abilities to apply techniques or teach them. We have one instructor here that suffered a bad electrical shock some years ago. This has left him with continual heart problems that sometimes mean he cant train for extended periods of time. His application of techniques in a randori environment though are almost without peer here in Australia. Would he perform better if he was fitter? Probably, but the reality is his performance now is better than many very fit people, even ones at his rank.
My personal definition of a martial artist is someone who is studying a martial art, and can perform the techniques required for the level of their grade, in a manner suitable for their grade, barring times of convalescence. Im not entirely sure what to do with accomplished martial artists that retire from the arts.