Trying to find birth records of James Lewis born 5th December 1862, possibly in Tyrone but definitely in dungannon from mid-1890s
This is the only James Lewis of about the right age I can find in Tyrone in the 1901 census. Is this your man? He was Presbyterian and born in Tyrone.
http://www.census.nationalarchives.ie/pages/1901/Tyrone/Derrygortrevy/Coltagequinn/1738828/
Statutory birth registration only started in Ireland on 1.1.1864 so if he was born before that you won’t find a birth certificate. You might find his baptism, but to do that we need to know his denomination. So anything else you can tell us about him may be helpful.