While the spelling variations can be insignificant, it seems that Emma and her likely father have both been recorded in official NSW records as HYETT, so across two generations.
Why? The John Hyett who arrived on the Grand Trianon in 1858 would appear likely to be the John Hyatt who died in McLeod's Creek in 1903. His death certificate states he had been in the colony for 45 years and the informant was his brother Richard. Similary there are several newspaper adverts/notices in 1863 from a James Burton trying to contact the John HYATT who arrived on the Grand Trianon in either 1858 or 1859. As far as I can see, only one John Hyett or Hyatt arrived at that time, although of course its possible that any other lists are just missing.
The Passenger Lists state that the John Hyett who arrived in 1858 aged 19 years was born in Middlesex. There are only two John Hyett or Hyatt's whose births were registered in Middlesex between 1838 and 1840. One is a John Hyett whose birth was registered Jun qtr 1838 in Westminster St Margaret Registration District. The other is John Hyatt whose birth was registered March qtr 1840 in Edmonton Registration District and who was baptised at Enfield St Andrew in Feb 1840. The latter is the John Hyatt who married Emma Birch (Emma Maria Hyatt's birth certificate confirms he was from Enfield) and who died in McLeod's Creek in 1903 and whose death certificate states he had been residing in New South Wales for 45 years.
I've not looked closely at what happened to the John Hyett born 1838 in Westminster, albeit that he would have been 20 in August 1858. Further research would be needed to eliminate him as the John Hyett who arrived in August 1858.
However, and I accept that John Hyatt born 1840 would only have been 18 years old and not 19 in August of 1858, there is other evidence to suggest that the Grand Trianon passenger was the Enfield John. Also travelling on the same ship was a Joshua Holland, who was John Hyatt born Enfield's first cousin. It seems likely they both emigrated together, following John's brother Richard who had arrived in the colony in May 1856 aboard the Phoebe Dunbar. Joshua's sister Sophia and brother in law Robert Haydon (who was also Joshua, John and Richard's second cousin) followed the others to New South Wales arriving in 1875.
Going back to your comment that Emma and her father John were more likely to be registered as Hyett. As you have obviously seen yourself on Emma Spruce nee Hyett's marriage certificate, she was illiterate and therefore its likely her surname was written as a literate person thought it was spelt. Herbert Spruce, who could write his name, listed her father's surname as Height on Emma's death certificate. Similarly, one of Richard Hyatt's granddaughter's surname was listed as Highatt on her marriage in 1920. And for that matter Maria Hyatt, Richard's daughter, is listed as Hiet on her marriage in 1889, her daughter Elizabeth's birth certificate in 1889 and her son Robert's birth certificate in 1896.
The same name variations appear in the records in England for the family too. I think the fact that Emma's marriage and the name on her children's birth certificates (of which the one I purchased and the one I found on ancestry both list it as Hyett) I really don't think is evidence that her birth would only be registered as Hyett.
Perhaps what is more of a mystery is why she is entered as Emily or Emily Maria on all of her children's birth registrations? Her marriage is as Emma and her death is registered as that, and hers and Herbert's death certificates confirm the birth registrations are the correct family. Even a later death certificate for one of her son's lists her as Emily. And its not just down to Herbert Spruce getting the information wrong. Herbert registered the birth of their son Herbert Harrington in 1904, he lists his wife as Emily Maria (there is a copy of that certificate on an ancestry public tree). Emma herself registered the birth of their son William Herbert in 1890 and obviously told the registrar (or whoever else registered the birth) that was her name. I have recently purchased a transcript of that certificate to see whether a different birthplace was listed for her, but, as with the 1904 certificate and her death certificate, it was listed as Tamworth.