It matters to you if you’re the original and then you are killed.
You are right that they are both an instance of person X but my argument is that this is not the equivalent to them being the same person in fact or even in law (whatever that means).
Also when/if this comes about I bet the law will side with me and define them as two different people in the eyes of the law. (And I’m not using this to fallaciously argue from authority, just pointing out I strongly believe I am correct—though willing to concede if there is ultimately some logical way to prove they are the same person.)
The reason is obvious. If they are the same person and one of them kills someone are both of them guilty? If one fathers a child, is the child the offspring of both of them?
Because of this I cannot agree beyond saying that the two different people are copies of person x. Even you are prepared to concede that they are different people to each other after the mental states begin to diverge so I can’t close the logical gap why you say they are the same person and not copies of the same person one being the original. You come partway to saying they are different people. Why not come all the way?
I think we’re on the same page from a logical perspective.
My guess is the perspective taken is that of physical science vs compsci.
My guess is a compsci perspective would tend to view the two individuals as being two instances of the class of individual X. The two class instances are logically equivalent exception for position.
The physical science perspective is that there are two bunches of matter near each other with the only thing differing being the position. Basically the same scenario as two electrons with the same spin state, momentum, energy etc but different positions. There’s no way to distinguish the two of them from physical properties but there are two of them not one.
Regardless, if you believe they are the same person then you go first through the teleportation device… ;->