Theoliver and Jake do appear to be the same person:
"United States World War I Draft Registration Cards, 1917-1918," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:K6CB-ZBW : 12 December 2014), Theoliver Jeter, 1917-1918; citing Caroline County, Virginia, United States, NARA microfilm publication M1509 (Washington D.C.: National Archives and Records Administration, n.d.); FHL microfilm 1,984,612.
"United States Census, 1940," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:VRYN-SVH : accessed 10 August 2017), Jake Jeter, Bowling Green Magisterial District, Caroline, Virginia, United States; citing enumeration district (ED) 17-3, sheet 2A, line 3, family 17, Sixteenth Census of the United States, 1940, NARA digital publication T627. Records of the Bureau of the Census, 1790 - 2007, RG 29. Washington, D.C.: National Archives and Records Administration, 2012, roll 4253.
"United States World War II Draft Registration Cards, 1942," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:V1KT-X9P : 7 April 2016), Theoliver Jeter, 1942; citing NARA microfilm publication M1936, M1937, M1939, M1951, M1962, M1964, M1986, M2090, and M2097 (Washington D.C.: National Archives and Records Administration, n.d.).
"Find A Grave Index," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:QK15-YZJN : 15 December 2015), Theoliver Jeter, ; Burial, Central Point, Caroline, Virginia, United States of America, Saint Stephens Baptist Church Cemetery; citing record ID 144830036, Find a Grave, http://www.findagrave.com.