Is opening the door of her house to her former teacher, who came to bid her farewell before his travel, and his attempt to hold her hand, considered a betrayal to her husband, and is she obligated to inform him of that?

Regret and seeking forgiveness are required for the sin you committed, but do not tell your husband about it. Cover yourself with Allah's veil (ستر الله); for telling him might lead to separation or loss of trust. It is sufficient for you to seek Allah's forgiveness and repent to Him, and do not despair of Allah's mercy. You can ask for his forgiveness in a general way without mentioning what happened.