Does divorce occur by moving the tongue with one of its words without opening the mouth, and what should be done in this case if one does not remember the word with which he moved his tongue?

The described situation is a case of waswasah (whispering of Satan/obsessive thoughts), and no divorce occurs thereby. Divorce does not take effect merely by doubting its occurrence, because the original state is the continuation of the marital bond until certainty of its dissolution is achieved. The marriage is established with certainty, and it is not removed by doubt. What is stated in the referenced fatwa (definition) does not apply to this situation. The questioner should hold onto his wife and stay away from thinking about divorce.