If you are looking anywhere other than
www.scotlandspeople.gov.uk you are wasting your time.
The deaths index there says that James Steedman, aged 83, mother's maiden surname White, died in Edinburgh in 1912. That, if it is accurate, would put his date of birth in 1828/1829. If you get a copy of it it will tell you the full names of both his parents.
There's a baptism of James Steedman, son of Henry Steedman and Mary Pitillo, in Edinburgh in 1823. (There isn't a birth certificate because there was no such thing until the start of civil registration in 1855.)
I see that James Steedman and Marion Campbell were married in 1854, and that they had James in 1855, Peter 1857, David 1860, Alexander 1862, Marion 1865, William Penicuik 1868, Mary 1870 and twins Margaret Campbell and Jane in 1873. The absence of a Henry among their five sons does suggest that perhaps James' father was not Henry, especially if Henry died in 1854, just before their first son was born. James' occuption varies from a stocker in a factory to railway packer to a porter.
In the 1861 census James said he was 30. In 1871 he said he was 38, born Greenside, Midlothian. In 1881 he was 52, and in 1891 he was 61. In 1901 he was 73 and in 1911 he was 82. So the range of birth years from the census, and from his death, is between 1827 and 1833. He's probably the James Steadman in the 1851 census with his grandmother Jean Steadman and Peter Steadman, 24. In 1841 the St*dman family consists of Peter, 67; Jean, 67; Peter, 25; Catherine, 25; Peter, 14; Alexander, 12; and James, 10.
I think the older couple are Peter Steadman and Jean Coalier, whose son Peter, born 1801, would have been recorded as aged 35 in 1841, when adults' ages are rounded down to the nearest 5 years.
Henry Steedman with wife Mary and children Henry, John, William, Thomas and Jane, are in the 1841 census together. In 1851 James Steedman, 26, is head of a household that includes his brothers Henry, John, William and Thomas and their sister Jane. James is a printer's pressman. I have not found him at all in 1861.
So to sum up I am certain that your James is not the son of Henry Steedman and Mary Patillo, and I think he is possibly the one who is a grandson to Peter Steedman and Jean Coalier.