I left my thoughts on the specific facts at hand in the edit notes. Some general thoughts:
Probably this discussion was not about copyright registrations, but rather PRO databases, which are run by private trade organizations (JASRAC, ASCAP, SOCAN, etc.) pretty much exclusively concerned with collecting licensing fees and dispersing royalties. They don’t really care about accurately tracking authorship details, these organizations (we hope, anyway) care about who should get paid.
This seems quite a bit of a different situation from the documents at hand, which are part of the public record maintained by the US Copyright Office, cataloguing all copyrights formally registered in the US.
The trouble with copyright registration records relevant to this edit, I think, is the difficulty in associating the records with any particular performance (which is what we most care about at MB). The registration applies to some specific published version(s) – copies of which are provided to the copyright office at the time of registration. Other versions can be registered separately (or not registered at all), and those other versions could have different authorship.