I'm assuming this is a transporter as exists in Star Trek, and not some kind of wormhole.
Imagine if it didn't deconstruct your original body, and only made the perfect copy at the exit. Would there be two "yous?" Under your definition, yes, but they are very clearly two separate entities. There is a "you" that walked into the entrance, and there is a "someone else" who walked out of the exit. I think a continuous consciousness is not only relevant, but crucial to a meaningful definition of "you."
And nobody post that "you die when you fall asleep" comic. It equivocates different definitions ofbghese words in a confusing and misleading way.