From what I understand, he would fit the absolutely most literal definition of the word “martyr”:
a person who is killed because of their religious or other beliefs.
(source)
Except that he wasn’t really killed for his religious beliefs. He was killed for committing crimes which angered the local population, the most significant of which was his burning down of the Nauvoo Expositor printing press in retribution for its printing of material exposing Joseph’s fraud. This was not a case of someone standing up for their religious beliefs and being killed because of it. This was a case of mob retribution against a conman who committed crimes. It was wrong for the mob to do this. It was also wrong for Joseph to burn down a printing press which painted him in a bad light. And getting in a gunfight would certainly not fit anyone’s definition of martyrdom. Had Joseph not burned down the printing press and not shot at his attackers, and if he had willingly accepted death rather than recant his beliefs, perhaps then he could be justifiably considered a martyr for his faith. But it’s not right for the Mormon church to call him a martyr when he really didn’t act like one.
Calling Joseph a martyr betrays the truth of what really happened and gives him a status he does not deserve.