I am not sure if this is the correct name. But I know its fairly common in the Charismatic movement. I have found this especially common in non-denominational churches. This weekend, I attended a Catholic Retreat which centered around some speakers telling their stories of their conversion experiences with the similar formula centered around feeling something was missing, turning their life over to Jesus and how they were so deeply changed. I find many times there is a lot of emotionalism centered around the stories.
I was wondering what others think of this charismatic practice. For myself, I get a little bit uncomfortable, as I am not an emotional type, and tend closer to the more aesthetic and philosophical aspect of religion. However, I have absolute respect for those who tend to the more charismatic movement.