Is this considered a khul‘ (divorce at the instance of the wife in return for a consideration payable by her) given that the husband is the one who desires the divorce and demands that his wife file a divorce case, and she does not want to remain in suspension and wishes to return to him what he gave her?

Khul' is the dissolution of the marital bond in exchange for a compensation paid by the wife, and it is an irrevocable divorce. It differs from divorce in that divorce is without compensation. If the husband divorces his wife before consummation, she is entitled to half of the Mahr (dowry), unless he had a private secluded meeting with her where intercourse was possible, in which case she is entitled to the full Mahr.

In the described situation, if the husband has already divorced the wife, she is entitled to half of the Mahr if he had not been secluded with her, and the full Mahr if he had been secluded with her. The husband has no right to demand more than that when documenting the divorce. If he has not yet divorced her, he can request Khul', and the wife can request either the completion of the marriage or a divorce.