I'm afraid that second number is unsupported. WW1 Facts gives 2.67 million volunteers and 2.28 million conscripted. Conscription In The First World War | WW1 FactsI got this from wiki but it appears the majority didn't volunteer to do their bit for King and Country.
Of these men, 2.67 million joined as Volunteers and 2.77 million as conscripts (although some volunteered after conscription was introduced and would most likely have been conscripted anyway)
Even your source, Wikipedia, supports those numbers (and thus disagrees with itself): British Army during World War I - Wikipedia and British Army First World War reserve brigades - Wikipedia. Note that your site explains that it gets its figure of 2.77 by assuming everyone recruited after conscription began was a conscript.
The UK parliament figures show approximately 2.5 million conscripted, but mentions that this includes those conscripted into 1920 (just as WW2 conscription didn't end in 1945). Conscription: the First World War
By contrast, Simpkins has figures of 2.47 million volunteers by the end of 1915.  Bourne gives figures of 1.19 million volunteering in 1914, and 1.28 million volunteering in 1915, agreeing with Simpkins's 2.47 million.  Spiers also agrees with those figures.  Note that these figures are to the end of 1915 and do not include either the volunteers who joined in early 1916 (before conscription), and nor those volunteering after that date.
Another Wiki page, Monthly recruiting figures for the British Army in the First World War - Wikipedia, has the monthly recruiting figures for August 1914 to October 1918, and shows 2.67 million volunteers and 2.28 million conscripts in that time, tying in with the WW1 Facts website. It also shows that just over 200,000 volunteered between January 1916 and October 1918, neatly filling the gap between the 2.67 million and the 2.47 million quoted by Simpkins et. al.The source is the official papers: General Annual Reports on the British Army (including the Territorial Force) for the Period from 1st October, 1913, to 30th September, 1919, so I'm happy to take them as accurate, short of pulling the papers from the National Archives which I'm not going to do.
Calculating the number of those conscripted against their will is problematic after March 1916 because there is no way to determine if the recruit was a willing or unwilling conscript. In addition, not everyone sent conscription papers was either finally recruited or served. Many were found to be medically unfit, others were placed in categories held in reserve only to be recruited should numbers fall below targets. There are also those who were conscripted but refused to join up. Still more would have been recruited but were too late to see action. This does not include all those who were exempt from conscription in the first place.
All figures in the millions rounded to two decimal places.
 Peter Simpkins, 'The Raising of the New Armies: Some Further Reflections', Britain Goes to War, p. 92.
 John Bourne, 'British Readiness for War', Britain Goes to War, p. 28
 Edward M Spiers, 'The National Response to the Outbreak of War, 1914', Britain Goes to War, p. 57.