I have to agree here. The "real" person is whoever Gatz/Gatsby happens to be at the time. Semiotically speaking, the name is relevant only in so much as to tell apart his past from his present. There has been a lot written, especially in the last few decades about the significance of roles in the function of identity. And it has also been said, in cultural theory, that since race is not the essence of a group's identity (culture is), that identity is existential: meaning, you create your own identity. Racial stereotypes are ways of grouping people together based on skin color: not based on their habits, culture, and history. And this applies to race, gender, even geography. In today's world, with access to information about all different cultures around the world, it is easier to create a myriad identity. So, identity, the real person, is a fluid thing. Gatsby is as real as Gatz was and vice versa.
In a sense, he is still that same boy, because that's the boy who fell in love with Daisy. So, he didn't disown his identity, he changed it and any future manifestation of his identity will be informed by new experiences and aspirations, but it will also be informed by the past.
Your question highlights the main theme of The Great Gatsby: illusion and reality. The answer to your question is probably, neither. Or, it might be, both. The point is that there may not be a "real," and if there is, readers don't know what it is.
First, we know very little about Jimmy Gatz. We don't know that he was an all-American boy, we don't know he was a hard worker, etc. Whatever Gatsby reveals to Nick about his past could be illusion, as could be the list that his father shows Nick. Notice that the father doesn't talk about how rigidly his son ever followed the schedule, just that he wrote it. So we don't know much about Jimmy. Keeping to the rigid schedule on the list could have been fantasy.
Concerning Gatsby, everything revealed about him is a carefully created persona, completely designed to win Daisy back. The reader doesn't know what any "real" Gatsby might be like. Not only does Gatsby reveal only his persona, but everything we get is filtered through Nick's eyes.
In addition to all this, Gatsby's dream of recapturing the past is illusion: Daisy never loved him in the same, all-encompassing way that he loves her.
Illusion dominates the play, and there's no way of knowing what any "real" Gatsby or Gatz might be like.
I want to think the person on the inside is Jimmy Gatz, but the guy that received the fulfillment of Wilson's frustration is indeed Jay Gatsby.
Jimmy Gatz fell in love with a girl as a teen and couldn't let go. I think he probably fell in love with being in love. He had dreams. He could go places. The book that his father brought with his list of resolves and plans demonstrated he had been a boy with the desires to do great things in his life.
I think sometimes we get caught up in things that drag us down too. The man, Jay Gatsby, got stuck. It might have seemed by all appearances to be an extraordinary life, but it hurt him in the long run. On the inside, we too, have dreams and resolves and desires and passions that run deep even when we don't have the capacity to pull them off. Our inabilities don't change our souls.
Gatsby may have had a soul change, but this is something we will never know. He certainly went to extremes. Being in an industry that likely caused deaths and being in an extra-marital relationship that caused deaths definitely shows us a side of a person that we have to wonder about. His actions demonstrate he gave up the American boy.
In my opinion, Jay Gatsby is the real underlying person. I do not think Jimmy Gatz has been around for a very long time.
To me, the last we know of Jimmy would be when he wrote that schedule in 1906 (Chapter 9). At that point, he still wants to work hard and improve himself.
But after that, he starts to want to take shortcuts. He doesn't want to work hard at college or work to put himself through college (Chapter 6). Then he sees Mr. Cody, and he goes off with him. At that point, he has given up the part of Jimmy that is honest and hard-working.
So, I do not really see Jimmy Gatz left anywhere in the time when the book is set -- only in the distant past.
Whether Gatsby likes it or not a piece Jimmy Gatz will always live inside him. The reader can see this time and time again with his love for Daisy. If it were not for Daisy I believe Gatsby would probably be war veteran comfortable home however this is not what happens. Since Daisy married for money I guess Gatsby probably thought that he could win her back with money. The reader can always tell that Gatsby could really care less for money. Gatsby does not even interact with the guest at his party which would increase networking and thus funding. Gatsby even threw chunks of colorful expensive cloths at Daisy.