The Zombie Stack Exchanges That Just Won't Die
Currently, the private-collection tag has exactly 1 usage, in the meaning of "home library," and is on the only question currently tagged home-libraries. At the moment, these are clearly intended to mean the same thing, and both are in usage, so one or the other is a candidate for being made into a synonym. The "-libraries" option would match other items in a standard taxonomy of library types that we're essentially implementing here, but private-collection might be slightly less ambiguous.
The slight ambiguity of home-libraries—private collections in the home vs. "your local (i.e., home) library"—can be easily overcome both by reference to the growing taxonomy (public-libraries, school-libraries, &c.) and more importantly by reference to the tag wiki (which I'm working on).
For this reason, and for those given in the questionelf, I think that private-collection should be synonym'd to home-libraries.
Perhaps only because I deal in antique data now, I think that "private collection" is a very different thing than any kind of library, including home library. However, this usage would be rather rare here.