This Illinois one is interesting. i had a quick look at baptisms close to his stated birth date. There are 2 in July 1862, neither with father John Watterson.

William Watterson
Date/Epithet 6 July 1862
Category Baptisms
Parish Maughold
Abode Water Street, Ramsey
Biography Parents: Thomas Watterson (car proprietor) & Jane Watterson (maiden name not stated). (MS09946/2/1)

William Edward Watterson
Date/Epithet 13 July 1862
Category Baptisms
Parish Arbory
Abode Douglas
Biography Parents: Thomas Watterson (mariner) & Jane Watterson (nee Clague)

There were 2 Wm Watterson baptised in 1863 - meaning they'd have waited more than a year which I don't think was typical.
William Henry Watterson
Date/Epithet 7 September 1863
Category Baptisms
Parish Malew
Abode Ballacam
Biography Parents: John Watterson (joiner) & Ann Watterson.

William Henry Watterson
Date/Epithet 7 September 1863
Category Baptisms
Parish Arbory
Abode Ballacrobin
BiographyParents: John Watterson (joiner) & Anne Watterson (nee Quayle) private bapt rec'd 13 Sept

I think for now this Wm in Illinois can be eliminated on the assumption his father is Thomas, not John. This is good, narrowing the field.

Thanks
Bev