There are more than two types, because the discussion so far has related to the certificates issued in England and Wales, and the ones issued in Scotland are different*. There are also Northern Irish certificates, and people born in Northern Ireland before 1922 have Irish birth certificates.
*Scottish birth certificates have always included the actual time of birth, not just the date, and in 1855 and since 1860 they also state the date and place of the parents' marriage (unless of course they weren't married).
As in England and Wales, the Registrar issues a short certificate which does not include the parents' details, and a full certificate with all information is available for a fee.