Is a divorce considered an irrevocable or a revocable divorce if the wife requested the divorce and relinquished her rights, and the husband divorced her without the intention of depriving her? Is it considered a divorce for compensation, and what is the considered intention?

If a husband divorces his wife in exchange for her relinquishing her rights, this is considered Khulʿ (divorce at the instance of the wife), and it is an irrevocable divorce (Talaq (definition) Ba'in) from which she cannot be taken back, because a divorce with compensation terminates the marital relationship. However, if he divorces her without compensation, it is a divorce (Talaq) and not Khulʿ. If it is the first or second divorce, he may take her back. But if it is the third divorce, she has become irrevocably separated from him (Ba'inunah Kubra), and she is not lawful for him until she marries another man and he consummates the marriage with her, and then he separates from her, as Allah Almighty says: "And if he has divorced her [for the third time], then she is not lawful to him afterward until she marries another husband."