It seems that if the Birth was not registered within 12 months, then the Birth needed the authority of the Registrar General to be registered, which is what you have stated.
A late registration would be done under the authority of the Registrar General, but that is also the wording used for a re-registration, which is the most likely scenario (under the Legitimacy Act) in this case, especially as
ShawThing seems to have identified a potential original registration in 1904.
A declaration would be a formal statement probably made by the parent/s?
The word "declaration" on an entry can have two very different meanings..... a "Statutory Declaration" is a statement made that would be related to a change or correction and would normally appear in a marginal note.
But in this case, "as per declaration" appearing in the informant column refers to the information having been given at a registration office outside of the district in which the event originally took place - quite understandable when it is 23 years after the birth and the parties may have moved some distance away ( the entry will still end up being made in the correct district though).
If the 1904 entry identified is correct, and a certificate is obtained from it, then it should have a marginal note rereferring to the re-registration in 1927 ( although occasionally that step seems to have been forgotten by the registrar when filling out the new entry).