This is an interesting question, because the only one I think we can discount completely from your list of four archetypes is the fall, as there is no sense in which Jerry's story can be compared to that of the original Adam and Eve story of the fall. If I had to pick one, I would probably pick the initiation, as there is a real sense in which the symbolism in this excellent story suggests that Jerry must pass "through the tunnel" as a kind of rite of passage to become an adult, and thus he fashions for himself his own challenge of initiation so that he can enter the adult world. Of course, there are elements of the task and the journey present in this story as well, as Jerry must train very hard and go through his own journey, both physically and metaphorically, as he goes through the tunnel.
Perhaps one of the most important ways that the initiation motif is shown in this story is through the way in which Jerry changes after he has been successful in going through the tunnel. Before, Jerry always had to exert his will to go to the wild bay and to practise their, however, note how this changes at the very end of the story when his mother advises him to take it easy and not to go swimming any more that day:
She was ready for a battle of wills, but he gave in at once. It was no longer of the least importance to go to the bay.
Now that Jerry has been successful and shown himself to be a "man" by going through the tunnel, he no longer feels the need to fight to go to the bay. That testing place is behind him now, symbolically indicating the shift and change he has gone through.