It is only possible to apply the literary criticism label "initiation story" to "The Open Boat" because some critics, not all, include adult experience as a class of initiation. The literary category of initiation story is drawn from the anthropological concept of rite of passage initiations, which move initiates from childhood at puberty to adulthood through indoctrination, maiming and enforced epiphany. Narratives based on similarities between coming of age and initiation rites are traditionally called coming of age stories or Bildungsroman.
If the critical literary category of initiation story is applied to "The Open Boat," we can say the correspondent is the initiate (the person being introduced to a new phase of life) because of the personal passage he makes during their struggle to survive on the open sea. The correspondent was a guest on the ship, a guest on the sea, while the cook, oiler and captain were in the fraternity of those initiated into the monstrous mystery of the sea: they know the life and death ways of the sea; he doesn't.
Through their journey to safety, the correspondent makes a passage of initiation from ignorance and innocence about the life and death nature of the fiercely impersonal sea to full knowledge of the struggle, on one hand, to befriend the sea and, on the other, to be victorious over the sea. His passage from innocence to knowledge comes about through watching the seamen suffer, be brave, be resolute and hope against hope.
It comes about mostly through his duty to good-naturedly and courteously row with or in alternation with the oiler; despite their exhaustion and fear, neither ever has a cross, begrudging, intolerant word for the other (except when the cook talks about pie):
The oiler plied the oars until ... the overpowering sleep blinded him. ... he touched a man in the bottom of the boat, and called his name, "Will you spell me for a little while?" he said, meekly.
"Sure, Billie," said the correspondent, awakening and dragging himself to a sitting position. ... the oiler, cuddling down to the sea-water at the cook's side, seemed to go to sleep instantly.
His passage of initiation comes about ultimately after his epiphany of finality when the oiler is carried, limp and lifeless, up the sand of their safety.
but a still and dripping shape was carried slowly up the beach, and the land's welcome for it could only be the different and sinister hospitality of the grave.
The corespondent's world view is forever altered as a result of his passage, his initiation into the monstrous mystery of the fiercely impersonal sea, the cousin of all nature, all equally fiercely impersonal, all able to kill the high or the low, the good or the bad with equal ease. These are the reasons why the correspondent is the initiate when this story is analyzed as an initiation story.
[Mordecai Marcus. "What Is an Initiation Story?" The Journal of Aesthetics and Art Criticism Vol. 19, No. 2 (Winter, 1960), pp. 221-228. Wiley Publishing.]