Here’s a marriage for Samuel Moore & Mary Murphy in 1861 which makes it fairly likely that you have the right parents for William John Moore:
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1861/09619/5488487.pdf
William John was born before the start of statutory birth registration (1864) so you need church records to find his baptism. Judging by the 1901 census he was Presbyterian. Do you know which church the family attended? Presbyterians in Ireland didn’t use the parish system and the church they attended was not always the nearest to where they lived.
http://www.census.nationalarchives.ie/pages/1901/Tyrone/Magheracreggan/Archill/1722491/
Most of the Presbyterian records for that area are not on-line but there are copies in PRONI in Belfast. If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
Samuel Moore in 1901:
http://www.census.nationalarchives.ie/pages/1901/Tyrone/Bomackatall/Ednashanlaght/1722682/
In 1826 the tithe applotment records list William Moor and partners farming in Ednashanlaght:
https://cotyroneireland.com/tithe/longfieldwest.html
Griffiths Valuation for 1859 lists 5 Moore farms in the townland. Samuel Moore was on plot 2 which was a combined total of about 13 acres. Next door on plot 3 was John Moore with a total of about 12 acres and next door again on plot 4 was Jane Moore (probably a widow) with just over 8. On plot 5 Hugh Moore had just over 2 acres and on plot 6 was Andrew Moore with just under 23.
Some probate abstracts (the wills themselves are in PRONI, save for Samuels’ which is on-line):
The Will (with one Codicil) of Samuel Moore late of Archill County Tyrone Farmer who died 5 January 1887 at same place was proved at Londonderry by David Watson (Junior) and Mathew Boyd both of Drumlegagh and Robert H. Moore of Drumclamph all in said County Farmers the Executors.
Moore Andrew of Ednashanlaght county Tyrone farmer died 23 December 1930 Probate Londonderry 30 January to James F. Gamble J.P. Effects £8.
Moore Margaret of Ednashanlaght county Tyrone widow died 14 February 1932 at Tyrone County Hospital Omagh Probate Londonderry 23 March to John James Love farmer. Effects £293 2s. 9d.
Moore William of Ednashanlaught county Tyrone farmer died 3 May 1935 Probate Londonderry 18 July to James F. Gamble farmer. Effects £29.
The Valuation Revision records for Archill show William Moore as being on plot 3a in the townland. That was a house and just under 18 acres, so presumably someone was farming it. William bought the freehold in 1912 under the Land Act.
According to the maps on the Griffiths Valuation site, plot 3a was on what is now the Magheracreggan Rd. It straddles the road not far from a forestry plantation. It’s fairly easy to identify on Griffiths maps.
http://www.askaboutireland.ie/griffith-valuation/index.xml
According to the 1911 census, William John’s house had 7 rooms, 4 windows at the front, a roof of tiles or slate, the walls were brick or stone and there were 6 outbuildings for animals and machinery.
I have no idea who lives there now (and there isn’t an easy way to find out). When you are next in the area you could knock on the door.
Can’t help with the relationship between William John & Beula. You would need to go back a couple of generations in the church baptism records, but they probably don’t go that far.
Elwyn