Is holding an unrelated man's hand considered zina (fornication/adultery), bearing in mind that the relationship is forbidden, and the intention was to complete the marriage, and the young man proposed marriage but the girl refused due to her family's rejection of her engagement during her studies?

Touching a non-mahram woman is a form of the fornication of the hand. The Hadith clarifies that Allah has decreed for the son of Adam his share of fornication. The fornication of the eye is looking, the fornication of the tongue is speaking, and the self wishes and desires, while the private parts either confirm or deny all of that. These matters are called fornication because they lead to actual fornication, but they do not entail the prescribed punishment (Hadd), rather discretionary punishment (Ta'zeer). The questioner must repent, stay away from this young man, and increase in seeking forgiveness and performing righteous deeds that expiate sins.