Does My Gaited Foxtrotter Need to Perform the Foxtrot? (frequently asked question)

When I bought my first Missouri Fox Trotter,  I was told to gather him up slightly to fox trot and gather up more for running walk. My "teacher" has me gather him up more than I had been originally, she says he has a running walk but no fox trot. How can I learn more about the difference in the gaits and the best way to achieve both. He is smooth when he moves which is my main concern after 3 surgeries to my spine. I sold a doll of a Tennessee Walking Horse because due to her pace riding became too painful.