They would think they are the original you, but I think the GP was saying that the original perspective would continue on the original consciousness-continuity/

They would think they are the original you, but I think the GP was saying that the original perspective would continue on the original consciousness-continuity/body/hardware. Cloning a hard drive can produce the same data, but without any networking, there's no reason for the original machine to know anything from the perspective of the new one