The concept in literary criticism of the initiation story is still a contentious one with many divergent ideas of what an initiation is or is not. Applying the critical label 'initiation story" to "The Open Boat" points out the contentiousness of the debate.
Initiation criticism originated with the anthropological classification of ritualized puberty rite of initiation passages from the childs' world into the adults' world. This passage is achieved though indoctrination, maiming and induced epiphany. The initiation story is said to have the same or similar elements as anthropological initiation rites. This sort of rite of passage story is traditionally called Bildungsroman or a coming of age story.
The difference between the the traditional Bildungsroman label and the newer critical initiation story label is that some critics include adult experiences of personal, social or cultural passage from one plane of understanding to a more encompassing and complex plane in the initiation story classification.
Having said this, the correspondent's experience in "The Open Boat" can, by some critical views, be analyzed as an initiation story: he is initiated into the life and death survival for existence of the sailor who must continually battle and try to befriend an impersonal and monstrous sea, as monstrous as the shark with glistening blue fin, body and tail.
There was a long, loud swishing astern of the boat, and a gleaming trail of phosphorescence, like blue flame, was furrowed on the black waters. It might have been made by a monstrous knife.
Two textual clues indicate through inference (indirectly, not directly) that the correspondent is the initiate (i.e., person receiving initiation) in this story. The first is where we expect such foundational information to be, in the exposition of the story. We are introduced to the cook of a ship, the oiler (overseer of fuel) of a ship, the injured captain of a ship and to a correspondent, evidently a passenger on the ship. It's a matter for induction as on intelligence tests: "Which one in this group does not belong?" Answer: D, the correspondent; he is the initiate.
The cook, oiler and captain are well initiated into the workings and threats of the fierce sea that is no respecter of persons: they know the sea and have battled against it before, though not like at present. The correspondent is from outside this initiated group; he is along as a guest upon the sea; he expects it to be friendly and accommodating to his goals. When the ship goes down, he changes from an uninitiated guest to the initiate in a fearsome rite of passage.
The second textual clue is later on when the correspondent wonders "why he was there." The narrator describes the initiated fraternity of the sea. All through the story, the oiler and correspondent bravely and good-naturedly exhaust themselves at the oar, rowing together or alternately: "the oiler and the correspondent rowed. And also they rowed." Now, the correspondent is described as one with the oiler, cook and captain in the initiated fraternity. In other words, he has succeeded in his indoctrination and torture and has become a man of the sea. This confirms that he is the initiate as there is no other way he might have become one with the other three.
They were a captain, an oiler, a cook, and a correspondent, and they were friends, ... [and] the correspondent, ... knew even at the time was the best experience of his life.
[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.]