Is the understanding that it is permissible to wear a ring if it is not specifically during the engagement period or if the suitor wears it for his fiancée correct, and why did the Prophet (peace be upon him) specifically ask for a ring as a dowry for the woman if it was not for the fiancée to wear or for the benefit of its value?

What you understood about wearing a ring without being restricted to the engagement period is correct. It is from the Sunnah for a Muslim to wear a silver ring. As for an iron ring, there is a difference of opinion regarding its permissibility, with the preponderant view being permissibility. The Prophet, peace and blessings be upon him, asked for an iron ring to find the least valuable thing that could be given as a dowry. The hadith clarified that the dowry can be little or much if both spouses agree to it, and this is the madhhab of the majority of scholars.