Is this divorce considered a third divorce, or a first divorce based on the husband's intention and his forgetfulness of previous divorces?

This divorce is considered the third, and with it, the wife becomes irrevocably separated from her husband and forbidden to him until she marries another man in a valid, desired marriage, not merely for analysis (tahlil), and he consummates the marriage with her. If this second husband divorces her, she then becomes permissible for her first husband. The fact that he does not remember the previous divorce is irrelevant, as what matters is his acknowledgment of the three divorces. 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. And if [the latter husband] divorces her, there is no blame upon them both if they return to each other and [are confident] that they can keep [within] the limits of Allah. These are the limits of Allah, which He makes clear to a people who know.}