IGI has the following refs, however they are all submitted, so unreliable, especially in this case as three of them seem to be the same person with differing info - which is why you are trying to sort it out, I suppose!!
Elizabeth SMITH, b 10 Jul 1797 Sydney, parents Thomas Smith, Ann Colpitts
Elizabeth Smith, b 1799 Hawkesbury; m abt 1828 John BAILEY, St Albans NSW; d 26 Mar, 1846
Elizabeth SMITH, b abt 1799 Macdonald River, parents Joseph SMITH and Mrs Joseph SMITH; d 26 Mar, 1846
Elizabeth Smith, b 1802 Hawkesbury, parents Joseph SMITH, Margaret HOLMES; d 26 Mar, 1846
Not much help, I'm afraid

From NSW BMDs
Marriage for Joseph SMITH and Margaret HOLMES at St Matthews Windsor, 1812; Ref:
V18121407 3A/1812
Also some births with parents listed as John BAILEY and Elizabeth (SMITH). There are more with mother just as Elizabeth without the SMITH mentioned. No districts given on these early ones.
V1832652 18/1832 John BAILEY
V18452082 31A/1845 John BAILEY
V18341086 18/1834 Thomas BAILEY
Judith