Unfortunately, people often put they'd been married ...years on 1911 census when no actual marriage ever took place, as various queries posted here can testify. Causes umpteen problems to those tracing their family history as you yourself have found.
Without goings into all the ins and outs at this stage, I can tell you that William Craigmyle seems to have given himself the name Fraser.
His parents were a George Craigmyle (bc.1843 Logie Coldstone, Aberdeenshire) who married (as Craigmile) Isabella Anderson on 2/6/1866 Old Machar, Aberdeenshire. They had the following children:
James b.3/2/1868 Tarland, Aberdeenshire
Betty (Elizabeth 1881 census, Bessie 1901 census) born 1871 Tarland
George b.25/1/1873 Lumphanan, Aberdeenshire
Ann (or Annie) b.1875 Tullynessle and Forbes, Aberdeenshire
Bella (or Isabella) b.1877 ditto
William Charles (??) born 4/6/1878 ditto
Joseph Anderson born 23/10/1881 Aboyne, Aberdeenshire
Think mother Isabella died 1885 Coull, Aberdeenshire as George is a widower in 1891 (at Aboyne) and 1901 (in Coull).
William (always as plain William) with family in 1881 and 1891 but the only entry I can find on 1901 census is a William F. Craigmilne aged 23, a Carter, in Leith, Midlothian (just outside Edinburgh) but he states born Alford, Aberdeenshire. Could this be him I wonder? Cannot find a person with this name before 1901, nor in birth records which makes me think it might be William Fraser Craigmile/Craigmyle.
Begs the question though as to why he'd 'adopted' the name Fraser as a second name in 1904 at Bessie's birth (and possibly by 1901 if the census entry above is him) as it doesn't seem to have been a family name as far as I can see.
As I said before, there is obviously a story here and just wish we knew what it was. Obviously, soon after Bessie's birth in 1904 the 3 of them moved to Newcastle where the family then became Fraser.
Just wish I could explain why.
Annette