I am sorry, I have to point out, that you definition of Messengers is not exactly as Quran teaches. Messengers of God are those Prophets who come with new Revelation and Holy Book. But those who are only Prophets, such as Soloman, or David, or, Aaron, are the promoters of religions. Qaim is a messenger, which means He is like Muhammad, or Jesus who brings a new Book. The Messengers are also prophets, but not every prophet is a Messenger of God. You cannot find a Messenger of God in Quran, who is not a prophet, but you find many prophets, who are not Messengers, such as the Bani Israel prophets (Anbiya)
Let's discuss this then, but I will open another topic for it. I propose it's the opposite.
Nabi means one who receives a revelation to God to be given to humanity. The meaning of revelation is very specific here, the type of khabar from God here, is meant to be safeguarded for all humans as well to a be proof of authority of whoever it gives and designates authority to. It's a trust that humans are meant to keep intact as revealed and not paraphrase in their own way or distort. As for Quran, as far as I know, is the only revelation that was guaranteed protection.
A Messenger means one who conveys God's Clear message to the minds of the people. The Ghadeer declaration is part of conveying the clear message with without the clear message would not be conveyed at all, but the actual words, are not found in the Quran.
Almost all Prophets were Messengers and almost all Messengers were Prophets. The exception to the rule happens once in a while. Once in a while a Prophet is not a Messenger because the message is firmly established and so he just reveals a divine to book to communicate from the creator and increase in guidance. Sometimes a Messenger is not a Prophet (as is the case with the Mahdi), because God out his wisdom doesn't want to reveal a book to safeguard, but just convey the clear message. Perhaps an example of that in the Quran is Talut/Saul.
We will discuss this another thread though.