Now to put the gray area filter onto this new info, if a photographer takes a picture of a family, does the photo belong to the family? Was the photo intended for the family? Or did the photographer, taking a family photo, expect that one member of the family should have that printed copy? This again falls into that fair use gray area.
Sorry but there is no grey area. These questions you are asking have been asked and answered by the law a long time ago.
The image copyright belongs to the photographer (unless they were in Canada or certain parts of Australia where the law was different).
The client purchased a print copy. That specific print belongs to the individual who contracted and paid for it (or whom the Photographer reasonably believed did so), not a potentially nebulous group referred to as "the family" which might grow or shrink or split over time and would therefore be a legal nightmare to define.
The owner can create a copy for archive purposes. They can lend, give away or even sell their specific print but if they do so they would need to surrender the archive copy as well. Except for the purpose of archiving they can not copy the print or distribute copies to anyone else unless the images are exempt from protection under some specific element of copyright law.
Whether or not these images are exempt as described by a previous poster will depend on the legal definition of "published". The term generally means wide spread distribution/made available to the public. It could well be that private commissions don't constitute "publication" and therefore don't require a copyright notice (meaning that they would not be public domain). This is a question I can't answer as I don't know the legal definition of publication or what happens with unpublished works.


