The 1983 date reflects the changes made in the nationality Act 1981. For issuing passports, birth registration is mainly of interest because of what it says about nationality. Identity is a separate matter: the question there is whether the person in the picture was the baby in the register or documents.
The GRO is now part of the passport office, of course, so they have direct computer access to the registration data. In fact there's an API for doing that, and its used by other departments too. Presumably the digital index we all use was first created for this purpose, and then extended back to records too old to relate to living people.
The internal instructions for passport issuers are on line, and refer to this (LEV):
If a customer does not send a birth certificate, we may still be able to check the birth record with the GRO (see Life Events Verification- checking birth records).
The Births and Deaths Registration Act 1953 specifies both a Short certificate of birth and a Certificate of registration of birth. The relevant sections are:
12 Certificate of registration of birth
At the time of registering the birth of any child, the registrar shall, if so required by the informant of the birth and upon payment to him by the informant of a fee not exceeding four pence, give to the informant a certificate under his hand in the prescribed form that he has registered the birth.
33 Short certificate of birth
(1) Any person shall, on payment of a fee of ninepence and on furnishing the prescribed particulars, be entitled to obtain from the Registrar General, a superintendent registrar or a registrar a short certificate of the birth of any person.
(2) Any such certificate shall be in the prescribed form and shall be compiled in the prescribed manner from the records and registers in the custody of the Registrar General, or from the registers in the custody of the superintendent registrar or registrar, as the case may be, and shall contain such particulars as may be prescribed:
Provided that any particulars prescribed in addition to name, surname, sex and date of birth shall not include any particulars relating to parentage or adoption contained in any such records or registers.
I think this 1953 act brought in the short certificate, though it's hard to be sure. The full certificate (or certified copy of an entry...) is not mentioned in it, so the previous legislation (probably the 1874 act) carried on. After 1953, most changes were by regulation not statute, e.g. in 1968, which is harder to keep track of.
By the way, it is now once again possible to use
the histpop.org site to look up laws (to 1939) and other stuff, as it's running properly! Someone must have bought it (or moved it to) a new server.