Just a few comments, perhaps obvious, but worth noting for completeness.
The burial date and age at death match the 18 Jun 1856 birth date you mentioned in your first post (assuming a two day lag from death to burial).That information undoubtedly came from the same informant who provided the information that you will find on the death certificate.There may well be additional information about parents on the death certificate.
That June 1856 birth date is actually consistent with the reported age of 53 in the 1910 census.That is how old he would have been on 15 April 1910, the date of record for that census.He would have been 53y, 9m, 28d old.
The June 1861 birth month reported in the 1900 is a little puzzling, but that is the same census where his parents were reported to have been born in England.It makes you wonder who was at home when the census taker called.
You have probably seen the passport application for William O's son William, listing William O's birthplace as Pittsfield, Maine.
I see several listings with ages for William O in the California voter registration lists 1900-1914 on Ancestry.com.I couldn't quickly match the listing to the year, but that might show a pattern that would reflect the shift between 1900 and 1910