- This raises a possibly ineffable question… The specialness that I’m claiming, and to which I’m alluding, is (that) consciousness provides an “identity” for the specific object that produces it; an identity that would distinguish that object from any other -- otherwise chemically identical -- object. If that sentence is coherent to you, do you accept it?...
I believe I understand what you're saying, yes. However, no, I do not accept it.
No personal sense of identity distinguishes the original from the copy. They both have an
identical personal sense of identity. As far as the copy is concerned,
it is you, and "you" are
other. The
only thing that distinguishes the two is location. Nothing about
consciousness is at play.
The copy and original are separate objects in separate places, and thus, are not just one object (and boy does that feel like a silly thing to have to spell out).

But each has
exactly the same "personal sense self."
Subjectively, each is you, and the other is not. From the outside, it depends on whether you imagine things from the original's perspective or the copy's. Neither one is
me, but they're both
a "you".
Imagine that you didn't know which was the copy. (The copying happened while you were unconscious or something.) How would you tell whether you were the original or the copy? You couldn't. But you would still
be you,
either way.
As for your case where "you" die before the new living copy is created: that is
indistinguishable from the case where a dead copy is provided and the original (still living) you is moved forward in time.
(As I mentioned earlier, English doesn't cope with this situation very well, which is why you're getting apparently contradictory answers from people. It depends on how you define "you", in a language where "you" (singular) normally only has one possible referent.)
So, bottom line, the copy would be a different object from the original, but that has
nothing to do with consciousness! There is
no "property of consciousness" that distinguishes the two! Only the fact that they are distinct physical objects. Like two loaves of banana bread.
(Interestingly, this is where you’re suggestion about space/time coordinates may threaten my little applecart… And, I’ll get to that as soon as possible.)
Can't wait.