Ann, Anne and Anna are all full names, not nicknames. "Annie" might be the name someone was given at birth or it might be a sort of nickname which has developed over the years. Generally, we refer to people by the name they use when they introduce themselves or use a name that they go on to tell us is acceptable to them.
I have a friend called Catherine. I have known her for over 20 years and she is "Cathy". That is what she calls herself and that is what everyone calls her. However, we used to work together and our boss insisted, for 3 years, on calling her "Cath". It infuriated her. She had never suggested to him that he could call her that (and she dislikes that particular shortening of her name). Her job application gave her name as Cathy. All the office documentation used Cathy. The boss's excuse was that he had a relative called Catherine who was always called Cath so he automatically used that abbreviation for all Catherines. Needless to say, my friend did not accept this and corrected him every time he tried to call her Cath.