Which of the protagonist's names in Jasmine by Bharati Mukherjee tells the true story of Jasmine's identity?
Your question seems rather odd until you realize that the premise of this novel, Jasmine by Bharati Mukherjee, is that each person who is connected to the protagonist also has a dramatic effect on her life and reflects that by the names they give her. In fact, each of the people she comes in contact with has a profound impact on who she becomes next throughout the course of her life.
The answer to your question is pretty subjective, as there are several people in her life who make her better and seem to kind of complete her; the one thing we can probably agree on is which of her names do not represent her true identity.
Jyoti (which means "light") Vijh is the protagonist's given name at birth. She is poor and destined to be a widow, alienated from all that is familiar, including her family. This seems like the most obvious choice of name which tells her true story, Jasmine is able to overcome this fate and therefore I contend that this is may be her truest self.
The protagonist marries a man named Prakash and her Indian husband calls her Jasmine. He is killed by terrorists, but before that he is an inspiration to his wife. He is educated and dreams of one day getting to America so he can finish his education, and he instills that hope for something better in his wife, as well.
Lillian Gordon is an influential woman in the protagonist's life, and she calls her Jazzy. Lillian is instrumental in Jasmine's rescue in several ways. First, she saves Jasmine from living on the streets; second, she is the one who gets Jasmine her first job in America.
Taylor Hayes is a fine man who treats Jasmine respectfully and cares for her on a long-term basis. He is a college professor for whom "Jase" (his name for her) works, and once he is free to care for her more deeply, he tracks her down and they go to California together.
Bud Ripplemeyer changes Jasmine's name to Jane. This relationship is healthy and loving, despite its difficulties, but it is not destined to be long-term for her.
In between all of these are others who mistreat and abuse Jyoti. She is often at the whim of her oppressors, so the names they give her and the person she is forced to be with them do not generally reflect her true self.
I would make the case that one of two possibilities exist which most reflect to the true character of the protagonist. The first is her original name, Jyoti. While she is not in a particularly good place for the first eighteen years of her life, she does still ultimately reflect the "light" for which she is named. As Jyoti she learns the resilience and strength of character she needs to draw upon during the rest of her life.
Second, the character of Jasmine is an accurate reflection of who she is. Her life with her husband, as long as it lasts, serves as the impetus for her to be something more. As Jasmine, she learns to yearn for better things. She gains a confidence and a desire for something better than what she has, and she spends the rest of the novel trying to achieve that goal.
It seems to me that both for the reasons listed above as well as for the fact that the novel is called Jasmine, Jasmine is the name which best represents who the protagonist is in this story.
The world is divided between those who stay and those who leave.
Jasmine is one who leaves, and she creates a successful life for herself despite sometimes unbearable oppression and heartbreak. She would not have done that as any other of her characters other than Jasmine.