Going back to square one...and it seems George Wm's mother died when he was about four years old so perhaps understandable to make an error with his mother's surname when marrying for the 2nd time in 1903. There is only one George William
CHURCH born in South Australia c1850.
Firstly I treat online trees with a great deal of caution. So many are simply copied from one another. A tree should be well referenced with all sources and if it were my tree I would attach all births deaths and marriage certificates that I have in my possession. My very extensive tree will never be placed online as already there is much misinformation on various websites and the authors refuse to accept my corrections [info from bdm certs] I always offer to share information including copies of any documents I have. Family history researchers with 'closed minds'

Climbing down from my soapbox now
http://freebmd.rootsweb.com/cgi/search.plMarriages Dec 1845
CHURCH Isaac Wellingbro' 15/843
CRAXTON Hannah Wellingboro 15/843
Births Mar 1847
CHURCH Thomas Craxton Wellingbro 15/403
CHURCH Isaac, Hanna nee
CRAXTON, son ?Thomas [died at sea] arrived in SA 16 Jul 1848 aboard the
SIBELLA from London via Plymouth.
http://www.slsa.sa.gov.au/fh/passengerlists/1848Sibella.htmAccording to this link Thomas, son of Isaac died on April 13th. There is diary of the journey on this link as well.
Arrival
http://www.rootschat.com/links/0jjx/Births in SA
CHURCH Mary Elizabeth
5 Nov 1848
Father Isaac CHURCH Mother Anna CRAXTON
At New Tiers Ade 2/97
CHURCH George William
30 Sep 1850
Father Isaac CHURCH Mother Hannah CRAXTON
At New Tiers Ade 3/115
CHURCH Isaac Charles
16 Sep 1852
Father Isaac CHURCH Mother Hannah CRAXTON
At Adelaide Ade 4/77
Death in Victoria
CHURCH Hannah
Father George CRACKSTON Mother Elizabeth
1854 Reg#2101
Isaac's 2nd marriage
CHURCH Isaac
DAWSON Jane
1855 Reg#2678
Births
CHURCH Robert
Father Isaac Mother Jane LOWES
At Woodford 1859 Reg#3361
CHURCH Thomas
Father Isaac Mother Jane DAWSON
At Terang 1861 Reg#21389
CHURCH Louisa Jane
Father Isaac Mother Jane LOWES
At Eddi 1865 Reg#2078
Deaths
CHURCH Robert
Father Isaac Mother Jane
5 years 1865 Reg#4878
CHURCH Thomas
Father Isaac Mother Jane
3 years 1865 Reg#4896
Deaths - Jane was obviously married to a
DAWSON prior to marrying Isaac.
CHURCH Jane
Father
LEURS Robert Mother Jane MURRAY
At Berwick 83 years 1906 Reg#4455
Buried at Berwick Cemetery
This must be Isaac's death as the Berwick cemetery has an Isaac buried in 1912
CHURCH Isaac
Parents unknown
At South Yarra 87 years 1912 Reg#7699
and death of son Isaac Charles born 1852
CHURCH Isaac
Father Isaac Church Mother Mary Anna Craxton
At Gembrook 78 years Reg#9282
As I mentioned before genealogy is not an exact science and many people use different given names. If you are so concerned with "Elizabeth Ernestina or Elizabeth Ernestina Jane" I suggest that you purchase some of her certs to validate the information already found. Perhaps Elizabeth Mary
BARR's death certificate may have accurate information. The info should possibly be 'born c1872' and even then who really knows the exact date if her birth wasn't registered. Possibly estimated from marriage and death certs....who knows!!
I cannot see a birth for a Jane
CHURCH to George W
CHURCH and Elizabeth Mary MEYER. I wonder if she was born before they married and is not registered.
You have not commented on the prison record for George
CHURCH son of Isaac
CHURCH. I have searched for the index entry but I understand there are volumes so fragile that they haven't been indexed. Marion BUTTON obviously had access to these volumes when she was compiling her own indexes at PROV.
I could find more but it's late for me now. I don't think there is any mystery

Cheers
Cando