So basically the idea of "eth ha-adam" is not specific to the Adam mentioned in Gen 2. Why mention it at all then? It does not prove Adam of Gen 2 is not Adam of Gen 1. I believe they are different people but "eth ha-adam" doesn't prove it.
In the past the claim was that "eth ha-adam" only appeared in Gen 2 and not Gen 1 and it was used to show the whole "8th day Adam" belief, until someone checked then that whole argument fell apart.
I agree… eth-ha-adam does not prove anything. The proof comes from ha-adam.