Is the husband's statement, "That's it, the relationship between us is over," considered a divorce, given that he did not utter the explicit word of divorce, and the wife admits her mistake and wishes to change, but feels that her husband does not desire that?

A man's statement to his wife: "The relationship between us has ended" is one of the allusive phrases of divorce (kinayat al-talaq (definition)). Divorce does not occur by it unless the husband intended it. If he intended divorce by it, then it occurs. If he intended something else, then divorce does not occur merely by the word.