So do I understand from that, that in the case of an adoption, the GRO are empowered to issue a re-registered (phony) birth certificate for an adopted child, giving it a new name and new parents?
They can issue a new birth certificate, but that is not as a result of any re-registration in the birth register.
Example: Sarah Smith is born. Her birth is registered in the usual way, naming her birth parent(s).
An adoption order is then made, and the child’s adoptive parents rename her Jane Jones. These details are entered into the Adopted Children Register. This does not affect the original birth registration, and is not searchable on FreeBMD regardless of date.
The adoptee can in due course obtain any or all of (1) a certificate based on her original birth registration (as Sarah Smith); (2) a short “birth” certificate based on her Adopted Children Register details (as Jane Jones, but making no reference to parentage or adoption); (3) an adoption certificate based on her Adopted Children Register details (as Jane Jones).
In your case, Marjorie D Shaw cannot be the same person as Barbara Hilton, because there are separate birth registrations for them so they are separate people. Adoption, as I mentioned earlier in the thread, does not generate a fresh birth registration.