It would seem that William and George Rayner, sons of John Rayner and Elizabeth Goldsmith also lived in Tasmania. William m. 1815, so the Rayner family is in Tasmania before the death of Robert Jones and death of mother Elizabeth Jones.
Elizabeth Jones is a very ordinary name and findingmarriage and death for Elizabeth (b 1798) might not be possible.
You might be able to locate her by association with siblings...witness at marriage of William or George in Tasmania, godmother of children baptised etc.
Tas BDM
William Rayner m Sophia Cullen 3 Jan 1815/168 Hbt 23/17yrs
Mary 5 Dec 1815/454 1816 Hbt
Eliza 23 Nov 1817/570 1817 Hbt
Sophia Christina 24 Sep 1819 /753 1819 Hbt
William James 16 Sep 1822/1769 1822 Hbt
Jessie Catherine 21 Oct 1822/2690 1828 Hbt
Rebekah 14 Jan 1830/3318 1830 Hbt
Henry Augustus 12 May 1834/5843 1834 New Norfolk
death Sophia Rayner 29 Apr 1835/4264 37yrs New Norfolk
http://portal.archives.tas.gov.au/menu.aspx?search=8A second marriage for William Rayner 1840, Mary Meney, Hbt, and death 1862 Castlemaine, Victoria, for William Rayner.
Newspaper item, 1862, has William, late Hbt, dies at home of son Henry in Victoria.
BDM TAS
Marriage for George Rayner b 1794, Hbt 1819, to Mary Pyers, death of George 1840, Hbt
I think you have said
"...All I have as a fact is she was in Tasmania living since 1818 and left Sydney after her father Robert Jones death with her mother Elizabeth Jones nee Rayner nee Goldsmith".
What does this refer to..where do you see Elizabeth Jones in Tasmania 1818?