Quick thing, everyone agrees that there is darkness in everyone's heart, yes? (Princesses of Heart exempt)
Now, here are my own speculations, after reading this whole topic. I like the memory idea, but think about it, don't Nobodies retain the memories of their former self? Of course, without a heart, these feelings are fake, right? But, either way, memories seem connected to the body, and not only the heart. Just a thought.
And, the name of this new enemy, Unbirth, this is where it starts. We all agree that there was darkness in hearts during the BBS timeframe. Although, there were no Heartless, as Xehanort hadn't found the door to the dark yet, and neither had he created artificial Heartless. But, with what did he create the first Heartless prototypes with? Possibly with knowledge of the Unbirths. Now, considering their names, Unbirths should essentially be something that has not been born. Or a reverse birth, perhaps. Perhaps, Unbirths are/were primary forms of the Heartless?
Stick with me here. Since Xehanort had not yet found this 'great power' mentioned in the Ansem Reports, maybe Heartless couldn't fully materialize? What if they tried, but couldn't, resulting in a reverse. In trying to be born, they are interupted by the fact that they CAN'T. So, they aren't born, but the darkness in the person's heart has already swallowed said person's heart, and what's left is an Unbirth--an incomplete Heartless.
This can further be justified by the fact that Unbirths do not exist past the point where Heartless come into the picture. So, possibly, after Xehanort finds this great power, Heartless can fully materialize, past the Unbirth stage. Although, I guess Unbirth is not a stage, as it is a complete failure of a Heartless trying to form.
So, my closing thoughts. Up until the point where Xehanort furthered his Heartless plans with this 'great power', Heartless could not complete their materialization, so each time a Heartless tried to surface, it would have to abort, and the remaining pieces of its attempt--that being the darkness in the heart, and possibly remaining light in the heart, as the Heartless can't completely swallow the heart with darkness, if it isn't created, right?--becomes an Unbirth. Basically, a failed, or incomplete Heartless. Or, if you will, a Heartless that was unborn.