When and why does James Gatz change his name in The Great Gatsby by F. Scott Fitzgerald?

We do not learn until Chapter VI of The Great Gatsby that Jay Gatsby was born James Gatz, although rumors suggest Gatsby is not his real name. His father called him Jimmy. He changed his name when he was seventeen years old. On the day he did this, he had been on the beach, not doing anything in particular, when he observed a yacht drop anchor on Lake Superior. He borrowed a rowboat and went out to warn the yacht's owner that this was a dangerous spot. When he introduced himself to Dan Cody, a wealthy and retired miner, he introduced himself as Jay Gatsby, a name Nick speculates Gatsby decided upon well before that day. Cody takes him in and mentors him, and Gatsby remains with him for five years. Gatz became Gatsby because he was reinventing himself and says the name "sprang from his Platonic conception of himself," a large part of the American dream (104). In his heart, Gatsby was no longer the child of "shiftless and unsuccessful farm people," digging for clams at the shore of the lake and fishing for salmon (104). He had a dream and a concept of himself that did not match the reality of his life, and "to this conception, he was faithful to the very end" (104).