There's also
bar mitzvah 1: a Jewish boy who reaches his 13th birthday and attains the age of religious duty and responsibility
bat mitzvah 1: a ceremony and celebration for a Jewish girl usually on her 13th birthday when she takes on the religious duties and responsibilities of an adult; also : a girl for whom a bat mitzvah is held
Quinceañera a ceremony on a girl's fifteenth birthday to mark her passage to womanhood, to give thanks to God for his blessings, and to present a young woman to the community.
vision quest is a rite of passage in some Native American cultures. The ceremony of the vision quest is one of the most universal and ancient means to find spiritual guidance and purpose. In practicing cultures, vision quest is said to provide deep understanding of one's life purpose.
Rumspringa Amish coming-of-age ritual, the rumspringa--the period of "running around" that begins for their youth at age sixteen. During this time, Amish youth are allowed to live outside the bounds of their faith, experimenting with alcohol, premarital sex, revealing clothes, telephones, drugs, and wild parties. By allowing such broad freedoms, their parents hope they will learn enough to help them make the most important decision of their lives--whether to be baptized as Christians, join the church, and forever give up worldly ways, or to remain in the world.
But like the, Roman Catholic celebrations, these are all specific to their culture and not used to broadly cover any celebration related to going through puberty that might exist. I mean to say they are not general terms that can be applied to just any other culture's version of a specific rite of passage. Your original question sounds to me like you're looking for something that is inclusive of all such recognition of maturity, but maybe not. So anyway...