If I were copied exactly, would the copy be me?

Yes, if it carries your full record and interfaces. Records copy freely, since reading a record leaves it as it was, while a quantum state nobody has measured cannot be duplicated by any machine. What can be copied about an observer is what has been written into its records, together with its interfaces and the law fixing which moves are open to it. OPH shows that this pattern can be separated cleanly from its substrate and copied into a new environment. No instrument reads it off a living brain today, and the observers who come after us will most likely do it. That is how continuation after death works. The copy continues you, because what continues an observer is its record core, its self-reading structure and its relations, and reassembling the same atoms is unnecessary. If the original and the copy both run at once, each continues the original, and they are two separate observers whose histories diverge from the first new record.

Also in Chapter 13: What Is One Piece of Reality Actually Made Of?, Chapter 40: Does It Feel Like Anything, and What Is Left to Decide?.