Is informing the teacher that a student cursed him considered backbiting, and what is the Sharīʿah's evaluation of the aforementioned student's actions?

Students must show respect to their teachers and avoid provoking them. Backbiting a teacher is forbidden. You did well in forbidding them and refuting the backbiting of the teacher. It is not permissible to inform the teacher of what happened, as this constitutes slandering. However, you can reprimand them by informing their parents or anyone with disciplinary authority if they persist. If they straighten up, then conceal their faults. You did well by not compromising on religious matters, for pleasing people by incurring Allah's wrath is fruitless, while displeasing them by gaining Allah's pleasure is a great good. This is based on the Prophet's (peace and blessings be upon him) saying: "Whoever seeks Allah's pleasure, even if it displeases people, Allah will suffice him against the people. And whoever seeks people's pleasure, even if it displeases Allah, Allah will entrust him to the people."