There's a lot of TV screens there, but only 6 "ones"; five before, and the sixth here. So we already know that the screens are multiple views of a smaller number of in-world avatars. Is it possible that the Architect built this screen room intentionally to make his own point, and to not muddy the waters he simply made all the Ones look like Neo? This sounds much more likely to me than Neo recurring as a person six times. Sure, if Neo figured it out, it would devalue the Architect's persuasion, but the path of the one builds a Messiah complex in The One; the One's ego wouldn't bother thinking about it.
The One isn't really a special person, though. The One is a single person to which The Matrix has attached all the accumulated rejections of previous people in The Matrix. There's nothing that special about Neo except that he's the one the Prime Program picked to attach these records to. The goal of the prime program is to convince the person these rejections were tied to to accept the Matrix, thereby balancing the internal logic to allow a reboot.
In other words, "The code you carry" is the sum total of Matrix rejections attached by the Matrix code to Neo, in the Matrix. When Neo accepts the Matrix, the Matrix can then balance the math. But that doesn't mean he's a program, any more than any other living human in the Matrix is "a program."
The key takeaway concept here is that the "problem" with the Matrix is that it doesn't handle rejection of the simulation; but humans are unpredictable. The Matrix "works around" this problem by collecting all the little errors caused by the rejections in one place, and attaching those errors to The One's avatar. By cleaning The One's avatar from the system, it also sweeps away all the collective errors, and the Matrix goes to 100%.
"The One" is then jacked back out into "The Real World" to restart Zion, as his avatar has been removed. Almost instantly, rejections of the Matrix begin again, restarting the cycle.
The reason Neo is special is because he rejects abandoning his avatar to the source; he continues being a rejection in the Matrix. This causes further instability, but his goal is to convince the Deus to reprogram the Matrix to be fault tolerant, and simply allow redpills to leave in peace. He convinces the Deus that this is the best way to both resolve Smith as well as to permanently fix The Matrix. When Smith merges with Neo, the Deus is able to finalize the Path... Neo's avatar is not only holding all the rejections of the Matrix, but also Smith's aberrant program; when Neo's avatar is "merged", so is all the errors. In the process, Neo dies, but the Deus is able to not only reboot the Matrix, but patch it to allow redpills to leave.