Is a woman forbidden (to her husband) by repeated divorce as mentioned, and is seeking Khul' from the second husband considered incitement?

The total number of divorces (talaqat) that have occurred to the wife is only two: the first was when he divorced her then took her back (raj'a), and the second was the divorce of prohibition (talqat at-tahrim), where the husband's intention, which was divorce, is the preponderant view. As for the subsequent divorce pronouncements, they did not count due to the absence of an intervening return (raj'a).

You may take her back, and you still have one divorce remaining. If you divorce her for the third time, she will become irrevocably separated from you (bainunah kubra), and she will not be lawful for you until she marries another husband in a marriage of desire, and he divorces her or dies.

It is not permissible for you to entice her to seek divorce or annulment from the second husband, as this is among the forbidden acts of incitement (takhbib), concerning which a stern warning has been issued: "He is not of us who incites a woman against her husband."

If a woman is divorced three times and marries with the intention of making herself lawful (tahlil) for the first husband, she does not become lawful for her first husband.