Is wearing rings during engagement part of the prophetic tradition, and should it be adhered to where the silver ring is for the man and the gold is for the woman, especially with the existence of a hadith from the Messenger ﷺ about a man who wanted to marry and asked him about an iron ring for the bride?

"Praise be to Allah and peace and blessings be upon the Messenger of Allah and his family and companions. As for what follows:

The ruling on exchanging rings during the marriage proposal has already been addressed in Fatwa (definition) No. 32384. As for the story of the mentioned Companion, it is found in the two Sahihs and it relates to dowry, not evidence of what the questioner mentioned. The story narrates that a woman offered herself to the Prophet ﷺ, and he asked a Companion who wished to marry her to find something for the dowry, even if it was a ring made of iron. Regarding wearing rings, it is permissible for a man to wear a silver ring, and for a woman to adorn herself with whatever she wishes from silver or gold."