Is a divorce that occurred in exchange for the wife absolving the husband of the deferred dower considered khulʿ (dissolution of marriage at the wife's instance), and is it permissible for the husband to remarry her based on the opinion of those who consider khulʿ to be a فسخ (annulment), knowing the jurisprudential disagreement on the issue?

Your divorce is Islamically valid because you explicitly pronounced it during the, Khul‘ (divorce at the instance of the wife). The jurisprudential dispute regarding Khul‘ without the pronouncement of divorce does not apply to your case. Therefore, your wife is unlawful for you and will not become lawful again unless she marries another man, consummates the marriage with him, and he then divorces her. The fact that the previous two divorces were not legally registered does not affect the Islamic ruling.