There does seem to be a contradiction.
It does say that Consular birth registrations do not confer British nationality
but it also says The advantages of a Consular birth registration are a British style birth certificate is available as evidence of British nationality.
Four years ago we registered our son and he has a British style birth certificate and this was used to obtain a British passport for him.
If the registration does not confer British nationality how do you prove that the child is British? Passport only?