My uncle managed to get copies of actual Orphanage records for 1902 and 1904 (attached). The 1904 record shows the birth date (month only, not day).
Matthew is register number 5 while my grandfather is number 6.
I have attached copies along with a history of the Olivet Home that may be of interest.
It is interesting to note the 1902 records show the "School" as opening in July 1902, but obviously both boys were resident in 1901 and the orphanage in some form was opened in about 1884 based on the history. Maybe there was a change of Administration in 1902.