There is no fundamental error it’s purposefully exactly as you state. Nagel is saying that consciousness is that second thing.
That's exactly what I'm saying is erroneous. Consciousness is the first thing, we are only led to believe it is a separate, second thing by a millenia-old legacy of dualism and certain built-in tendencies of mind.
I doubt Nagel would go out of his way to offer such an unnatural linguistic construction, and other philosophers would adopt this construction as a standard point of reference, if that was the sole intent.
>So then are you saying there is no such thing as consciousness?
No, not at all. I'm only saying that if we want to talk about "the consciousness of a bat", we should talk about it directly, and not invent (implicitly) a second concept that is in some senses distinct from it, and in some sense comparable to it.