Dudley Registry Office is trying so hard to help me find my William Taylor

We were hoping to find the one on the Quinton School Register with birthdate listed as 8th Feb 1878...parent/guardian Elizabeth Taylor...occupation nailer..resident Quinton
... but so far they can find none with the birthdate I had suggested , or with Joseph as the father.
I received another email from them today..... re the Quinton School register and noticed that there was another William Taylor on there ( born 20th Feb 1882)
This one's mother is named as Elizabeth, a nailer and resident in Quinton, but the father is Thomas Taylor. On the School Register parent/guardian is listed as Thomas Taylor resident in Crock Street....and there are 2 other children on there ( John born 12th Feb 1879 and Elizabeth born 25th Jul 1885)
the following marriage would fit...
1877...TAYLOR Thomas.... WOOD Elizabeth ....Halesowen, St John the Baptist .....Dudley Register Office 014/004/314
( I have posted this for lookup)
I think this is the family on the 1881 census
Hagley Road, Ridgacre, Worcestershire, England
Thomas Taylor (30) potter, wife Elizabeth(22) and son John(2) all born Quinton
and in 1891....Crock Street, Hill, Worcestershire, England
Thomas Taylor(39) flower pot maker, wife Elizabeth(31), sons John(12) & William(9) and daughter Elizabeth(5) all born Quinton
and 1901...New England, Hill, Worcestershire, England
Thomas Taylor(49) road mender, wife Elizabeth(41), son William(19) single..labourer in tube works, daughter Elizabeth(15), son Thomas(9) all born Hill, Worcs., lodger Henry Shrettch? (61) widow..labourer on rd. born Halesowen
As the son William is still living with the family ( and still single) in 1901 this can't be my William Taylor... as I already have found my great grandparents in 1901 living in Langley.
BUT... I am wondering if there is some connection between these Quinton Taylors and my William.
Going back to that school register ....(I have listed entries below )
...the William Taylor born in 1878 was admitted to the school in Sept 1883 ( so after the 1881 census)...
Is the Elizabeth Taylor ( nailer) the same Elizabeth that is the mother of the Taylor children with father as Thomas?
Could she be his Aunt? or some other relative acting as guardian .... did something happen to this William's parents and he moved in with the family after that 1881 census?
If so, he is not living with them in 1891.
I suppose another possibility is that this Elizabeth Taylor is a single mum .. could she be related to John Taylor?
498 1883 9 Taylor William 8 2 1878 Elizabeth Taylor Nailer Quinton
525 1884 3 Taylor John 12 2 1879 Thomas Taylor Labourer Crock Street
680 1887 6 Taylor William 20 2 1882 Thomas Taylor Labourer Crock Street
884 1891 4 Taylor Elizabeth 25 7 1885 Thomas Taylor Labourer Crock Street
974 1893 4 Taylor Annie 24 11 1886 William Taylor Labourer Quinton
975 1893 4 Taylor Florence Eliz 18 1 1889 William Taylor Labourer Quinton
I am thinking that I will apply to GRO for a birth certficate for William Taylor born on 8th Feb 1878, father Joseph ( not quite sure what to put down for 'place' though... should I put Quinton or Worcs/Staffs?) ..... this may clarify things if they can find one

In the meantime... I am trying to see if I can find any Joseph Taylors associated with this Quinton family .... did John Taylor have a brother Joseph? or a sister Elizabeth?
Any help trying to sort out this greatly appreciated

Expat