So--am I right to suspect that this 'birth diploma' isn't really her full official birth certificate and I should probably go ahead and order a formal copy?>
Yes. This is not the official birth record. I was born in Elizabeth and have a certificate of birth (with seal and all) which I believe was issued by the hospital where I was born. I guess this was the practice in the State of New Jersey. When I needed my birth certificate for marriage (out of state), the certificate was not accepted. I needed to present the full birth record issued by the city. Avrohom Krauss