Hi there,
I am becoming confused. Using the quality research in the replies, I can join the dots from the birth in South Australia, the marriage in South Australia, through to the death in New Zealand for Elizabeth. I am confused as to where the conflicts are, as I don't see any significant ones along that line. Yes, there's differences in the spelling of her surname. However, that is not at all unusual.
If I were searching for an Elizabeth COIL in South Australia, I would not doubt a birth registered as Elizabeth daughter of Joseph COYLE, nor would I be concerned about a confusion between Daniel and Joseph, particularly if Elizabeth's dad may not have been around when she was growing up.
Perhaps in 1910 the Reverend did not ask her how to spell her surname, perhaps she had not been given the opportunity to check the parish register where the Reverend recorded it. Afterall, the Reverend would have interviewed the couple in the weeks before the ceremony. He would have been required to ask for proof that an adult, responsible for Elizabeth, was providing consent for her marriage. The index clearly notes that she was aged 16. Perhaps she was not a great speller herself.
The marriage laws, both church and civil, required her to be 21 years of age before she could give her own consent to her own marriage, unless she had been previously married, (and therefore became qualified to give her own consent, regardless of her age). If this was a marriage in New South Wales, then I would comment "NSW marriage act required the consent for a marriage where either or both were not yet 21 years of age. The clergy recorded this on the original parish register in the 'white space' near where the pre-printed words are "according to the rites of" and the clergy records the particulars of those rites, AND then should write down "Written consent for Elizabeth, a minor, was given to this marriage by ..................., (both given and surname), .............. of the bride (father/mother/guardian/ etc as in their relationship to the bride). Many clergy may have written "Consent for ...... rather than "Written consent for ....." But NSW Marriage Act required consent to be noted.
I do not know South Australia's Marriage Act at that time, but I am most surprised that it would seem that consent was not recorded on your copy of the marriage certificate. I am wondering if you are holding the ceremonial document presented to the bride at the time of the marriage, rather than the actual civil registration of the marriage. It is my understanding that South Australian BDM certificates hold scant information, so I am not suggesting you spend funds on ordering a new set of records. Simply put, I find it unusual that a marriage certificate for a 16 year old lass does not note who gave consent for her to marry.
Whether her father was Daniel or Joseph or Daniel Joseph or Joseph Daniel, or COIL or COYLE or other variations seems much less significant to me. You see, to me, COIL or COYLE are simply variations which are more likely due to the recorder's literacy levels rather than the Bride's literacy levels. The information was informant driven. In that era, the clergy were significant members of the community.
I am not sure where the 'gaps' are, except that like most of us, you would like to record the name of the ship that carried Elizabeth to New Zealand. I am very doubtful if you will achieve this goal. Within most of the family history groups based in Australia I am quite sure that none would expect to achieve that goal with complete certainty simply because New Zealand and each of the six colonies that federated in 1901 to form Australia were all still British Colonies, in an era before passports. The paperwork that the shipping companies did create often did not note the given names of their passengers, nor even the gender of any children. So, it is unlikely to find Elizabeth travelling as Mrs Elizabeth GREEN. Perhaps they were recorded as Mr and Mrs S GREEN and child/ren or Mr and Mrs GREEN + infant. Perhaps they purchased their tickets in the hour before the ship departed, when the clerks were perhaps at their own busiest time of the voyage. But basically, the requirement for paperwork with 'full details' was not a high priority for the shipping companies or for the governments of either NZ or any of the states of Australia. In 1910, they needed to know that the fares had been paid for, the passengers were on the correct deck for the fare paid. It was simply a short voyage between British colonies, with no foreign ports in between.
All speculation, but basically I am not spotting any gaps/conflicts, and I am one who has taken a lot of criticism over many years for being 'too strict' some would say that I am far far far too "pedantic".
Cheers, JM