Well having read the seventh book several times, my understanding of the situation is this:
* There were two souls within <PERSON>; his own pure and complete and a part of <PERSON>'s soul that latched onto <PERSON> when he survived <PERSON>'s Killing Curse as a baby.
* As long as that soul stayed in <PERSON>, <PERSON> would not die.
* In the forest, <PERSON> directed a Killing Curse at <PERSON>, without knowing about the part of his soul in him.
* The Killing Curse (<PERSON>) kills the victim by destroying his/her soul. Hence, there is no other fatal injury caused by the curse. Just a bruise where the curse hits the victim.
* So when the Killing Curse hits <PERSON>, its a choice between which soul survives and which gets destroyed, <PERSON>'s pure unblemished one, or <PERSON>'s fragment.
* If <PERSON> had chosen to move on, I believe his body would still be alive in the form of <PERSON>'s unintended Horcrux. So there would be two <PERSON>'s walking and talking, except one would be using (and in complete control of) <PERSON>'s body.
But as @cst1992 mentioned in the comments to the question, there is also the matter of the Elder Wand, which cannot kill <PERSON>. Obviously the wand recognized that there was a fragment of another soul, apart from his own, within <PERSON>'s body and would have killed only that fragment. That choice between the two souls would only happen if <PERSON> had used any other wand, apart from the Elder Wand.
There's also this business of <PERSON> being tethered to life as long as <PERSON> lives since <PERSON> took <PERSON>'s blood during his resurrection to a full-bodied being.
So technically, <PERSON> couldn't have "moved on" at all, since his soul was not harmed (and could not be harmed) in way at all.
I believe, in King's Cross, <PERSON>'s mind was clearing itself of <PERSON>'s soul and was able to think and reason clearly, without any worry or guilt or emotion to cloud his thoughts. 773
There is no real new information which comes out. It's just <PERSON> deducing everything for himself, with his image of <PERSON> helping him out. Which is why <PERSON> says :
“My dear boy, I have no idea. This is, as they say, your party.”
Of course, as to the existence of a "King's Cross",
“I think,” said <PERSON>, ”that if you choose to return, there is a chance that he may be finished for good. I cannot promise it. But I know this, <PERSON>, that you have less to fear from returning here than he does.”
When <PERSON> talks about "returning here", maybe <PERSON> intended to show all magical folk go to their own "King's Cross" upon their deathbed.
P.S: Maybe we Muggles go there as well. There's no way of finding out for real of course! 773