How can one reconcile the fatwa that a vow made in jest is binding—even if not intended—with your fatwa that a vow is not binding if its establishment was not intended?

The Prophetic Hadith: "Three matters, their earnestness is earnest, and their jesting is earnest: marriage, divorce, and return (to a divorced wife)," indicates that these matters are not invalidated by jest or मजाक. Scholars have analogized to this the rights of Allah, such as vows (nathr). Ibn Taymiyyah held that these actions involve a right of Allah Almighty, and therefore it is not permissible for a servant to jest with them. For this reason, legal rulings have been established for them even if they are issued in jest. As for actions related to the rights of individuals, such as selling, jesting is permissible in them.

In your situation, your recounting of the vow with the aim of ascertaining its matter is not considered a serious vow, because scholars have differentiated between one who initiates a vow and one who recounts it. Indeed, whoever recounts the word of divorce—which is more severe than a vow—divorce does not fall upon him, because he did not intend its meaning but rather the recounting or teaching.