I've now found, through my subscription with World Vital Records, what should be absolute proof for you of certain things anyway. There is a familey history published in 1905 called "The Turnleys" by Parmenas Taylor Turnley (your Milton's grandfather!), so it should be relatively accurate. On pages 234 and 235 of this book, he says that his daughter Mary (called Mamie) was born 7 July 1858 in Pottstown, Pa and died 24 Nov 1880 at her residence at 303 Chestnet Strett in Chicago leaving an infant boy just 3 weeks old. Mary married Milton C. Lightner at her father's residence at 352 LaSalle Avenue in Chicago on 24th Sep 1879. Milton C. Lightner died at his father-in-law's residence, Highland Park, of typhoid fever on 13 Dec 1895.
Nowhere in the book does he give Milton Sr. the middle name of Clarkson, just Milton C. Lightner. Now that you have his death date, you should be able to get all this information from his death certificate of 13 Dec 1895. There is another book, called the "Ryerson Geneology" which gives Milton Sr.'s birthdate as 30 Dec 1850, so that fits the 1880 M. C. Leitner who is 29 in the 1880 census (I also rechecked the 1880 census image at Ancestry and indeed the address of MC and MT Leitner is 303 Chestnut Street, so that now becomes a factual source.
If the "C" stands for "Clarkson", then maybe he was named after the reverend out of respect. As to the parents of Milton C. Lightner, born Dec 30 1850, that might be on the death registration. He won't be in the 1850 census, just in the 1860, 1870 and 1880.