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While looking for information about a name of the ceremony Native Americans did to become a man?
Indian country had well over Five thousand tribes before first contact and each had their culture and ceremonies, many ceremonies were basically the same because all were a form of prayer to and from Creator.
Remembering that (Religion in Indian country) was Fear (Love and Obey) Creator!
Some tribes send their young man into the wilderness for a period of time to learn about Mother Nature before becoming a man.
As a Native American of the east coast, I look at this question as the old Hollywood black and white films with an Italian dressed as an Indian Medicine man dancing around with a teenage boy that just came back from battle as some kind of a hero?
I use an Italian simple because film directors back then used dark skinned Italians in movies as Indians?
They could not find a Native American?
In my Native American Tribes we use real Traditional American Natives!
Gwaitoko (boy to man ceremony)
As boring as this answer is going to be the truth is that as soon as a boy in the culture in our tribe turns thirteen, a Four Corner ceremony in a sacred circle and they are excepted to be a man.
By the way, if you are seeing a Medicine man other than by accident, he is not a Native American Medicine man from an East coast tribe, this goes against Creators blessings! Ceremony to become a native man?