I'm thinking that a martial student becomes a martial instructor in much the same way that a high school student becomes a high school teacher. Go through the classes; graduate; teach students.
That may describe how one becomes qualified to become an instructor, but to actually become an instructor is like getting a job. You have to interview for the job, there needs to be a job opening or you need to create the job opening, and you need to get selected and accept the job.
Anyway, being an instructor is like a job. If you aren't qualified, you either won't get the job or you will not do well at the job should you get it.
At what point does one make the transformation from a martial student to a martial artist?
I don't believe there is a transformation. Martial artists are students too.
IMO, it is more about priorities. For instance, a mark of a martial artist is...
1) When one does not care as much anymore about what others think of them and instead cares more about improving themselves to be the best they can.
2) When one does not care as much about being just like someone else, but instead has the courage to be themselves learning and testing their own strengths and weaknesses.
3) When rank is not as important as a measure of one's worth, but instead as a sign of hardwork and training.
4) When ego is not as important as continuing to train and live martial arts.
IMO, it is about what priorities people have and follow-through on that differentiates someone just in it for the ride and someone in it for the thick and thin. Some would say it is all about commitment.