Both, but only one of me at a time

Both copies are "me", for all intents and purposes...althoguh as soon as they differ (within microseconds after whatever force ensures they are identical stops) they each begin to diverge, and become a seperate entity.
I'm not claiming that a discontinuity in consciousness makes the copy no longer "you". I'm claiming that the thought experiment of dismantling you and constructing an identical copy somewhere else is not the same thing as moving you to that place (which is demonstrated by missing out the step of dismantling you).
Yes, there's a difference between moving something and copying it...but the question is which one is "really" you. Both copies are still you. The only way either of you can detect any difference is by history, essentially. There's no physical difference (except your position). Assume (for the sake of argument) you don't know exactly how a transporter works. You don't know if it destroys you, moves you, or whatever. After it runs there's a copy of you in both places. How do you tell which one is the original? How do you determine the one on the other end is a copy, and the one on this end is the original? There's no physical, detectable difference except in position.
The reason to restrict historical knowledge is to show that the distinction between the "real" you and the "copy" is entirely arbitrary. It rests on a distinction of spatial coordinates, which makes as much sense as assuming you are a different person when you walk across the room.
Imagine a transporter that moves one particle at a time to the new location, and replaces that particle in you with one pulled from a stock of basic elements (ignore the physics limitation of the uncertainty principle...it's a thought experiment after all). At what point do you stop being you and start being a copy?
The whole point is that "you", as an entity seperate from your physical form, is a myth. All of this about "but I won't see anythign there" is a red herring that assumes you are something more than the product of your physical makeup. IF the original is destroyed, it's no longer "you"...there's no "you" there anymore. But the exact copy is "you" as much as the original is "you". The orignal no longer exists. If neither are destroyed, each is just as much "you" as the other, with the same thoughts, memories, likes, dislikes, and so forth. Destorying the original and reconstructing it, assuming exact copies, is exactly identical to mvoing you. There's no functional, practical, or detectable difference involved. The orignal "you" wouldn't detect anything, either (there is no "original" you after the destruction). But "you" have been recreated in the copy, with the exact same everything. So "you" still live on.
I really don't like the transporter thought experiments, they seem too prone to spiral off into details and side issues. I prefer thinking about it in terms of the robotics experiment:
Imagine that you can replace part of your body or brain with an idetnical cybermetic copy. This cybernetic piece exactly replicates the functions of the part of the brain or body it replaces. You can replace pieces in as large or small a chunk as you like.
SO, questions related to this:
1. If you replace the body, but leave the brain alone, are you still you?
2. If you replace the brain, but leave the body alone, are you still you?
3. IF you replace a part of the brain, say 1%, are you still you? If yes, what if you replace 10%? 50%? at what point do you stop being you?
4. Same as 3, but the body.
5. If you replace your brain a piece at a time, say 1% per week, does that change the answer to 3? Same for 4?
YOu can apply these same questions to a transporter, or a transporter-like idea. Assume the transporter works by creating the exact copy of you on the other side of the room from available materials, then over time slowly swaps out a small portion of the original you with a small portion of the second you. At what percentage of swapping out are you no longer you? Do you ever move to the second location?
Personally, there's no functional difference between moving me, and deconstructing and reconstructing me at the end location. There's no possible physical way to determine the difference (assuming no glitches). Either one is "me", and I continue on as long as at least one copy exists. If both copies exist, they are both as much me as anything else, with neither having more right to the claim of me than the other (except by arbitrary standards).