I think it's misspelled on Jesse death record. On the records on his siblings - his father is William H(enry) Craig and mother is 'Mosier'. The birth record here is not proof as the marriage record doesn't give parent names. Maybe you can get the actual marriage cert to see if there is more. From census data it appears that Wm H - was married to Martha Salmons/Sammons prior to marrying Louisa. You can search all the individual children on familysearch,
Kentucky, Marriages, 1785-1979
https://familysearch.org/pal:/MM9.1.1/FWYC-1NT
groom: William H. Craig
bride's name: Louisa Mosier
marriage 13 Dec 1873 Grayson, Carter, KY
Virginia, Births and Christenings, 1853-1917 Possible mother of Jesse ???
https://familysearch.org/pal:/MM9.1.1/X5LG-WYM
name: Louisa Mosier
gender: Female
birth date: 02 Dec 1853
birthplace: Scott Co., Virginia
father (or slave owner): B.L. Mosier
mother's name: Susannah Mosier
Ohio, Deaths, 1908-1953
https://familysearch.org/pal:/MM9.1.1/XZF2-V4R
Susan Jane Rose
Death 18 Oct 1937 Bloom, Scioto, Ohio
residence: South Webster, Ohio
street address: R #1
Female age: 52y 11m Married white
birth 18 Nov 1884 Ky.
burial date: 22 Oct 1937 So. Webster
father: Henry Craig birthplace: Va.
mother: Louise Moisier birthplace: Va.
spouse: John Rose
Kentucky, Death Records, 1911-1955
https://familysearch.org/pal:/MM9.1.1/NSWX-PNR
Bernnard Craig
Death date: 1945 Sophie, Carter, KY
father: Henry Craig
mother: Lou Mosier
spouse: Myrtle Craig
US Census 1880 Precinct 6, Carter, KY
https://familysearch.org/pal:/MM9.1.1/MCCH-J18
self Wm. H. Craig M 39 KY abt 1841
wife Louisa Craig F 24 VA abt 1856
son Wm. F. Craig M 16 KY
dau Nancy Craig F 14 KY
dau Josaphine Craig F 12 KY
dau Elizabeth Craig F 10 KY
dau Pheba Craig F 5 KY
dau Sarah Craig F 3 KY
son B. M. Craig M 1 KY
U S Census, 1900 ED 33 Magisterial District 6, Maddox Precinct 9, Carter, KY
https://familysearch.org/pal:/MM9.1.1/M9WP-4XQ
head William H Craig M 60 KY Jul 1840
wife Louisa Craig F 45 VA Dec 1855
son Bernard M Craig M 21 KY
son Jesse J Craig M 20 KY
dau Ollie Craig F 16 KY
dau Rosa B Craig F 14 KY
dau Stella Craig F 11 KY
son David L Craig M 9 KY
dau Martha M Craig F 7 KY
dau Doria Craig F 5 KY
dau Sarah Rose F 24 KY
granddau Hattie M Rose F 12 KY
grandson Eddie E Rose M 1 KY