It appears to be the same person - his compiled military service records would say for sure.
Moses Beck
Regiment Name Snider's Battalion, Missouri Cavalry
Side Confederate
Company D
Soldier's Rank_In 1 Lieutenant
Soldier's Rank_Out 1 Lieutenant
Film Number M380 roll 1
Moses Beck
Regiment Name Capt. Beck's Company, Missouri
Side Confederate
Company
Soldier's Rank_In Captain
Soldier's Rank_Out Captain
Film Number M380 roll 1
American Civil War Soldiers
about Moses Beck
Name: Moses Beck
Side Served: Confederacy
State Served: Missouri
Service Record: Enlisted as a Private.
Enlisted in Company D, Snider's Battn Cavalry Regiment Missouri.
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1850 United States Federal Census
Name: Moses G Beck
Age: 18
Estimated Birth Year: abt 1832
Birth Place: Kentucky
Gender: Male
Home in 1850(City,County,State): Hartford, Pike, Missouri
Amos Beck 46 Clergyman Regular Baptist b. KY
Nancy Beck 37 wife b. KY
Moses G. Beck 18 [can't make it out - looks like "To The muses"] b. KY
Isaac F. Beck 15 farmer b. MO
Joseph J. Beck 12 b. MO
John P. Beck 10 b. MO
Mary Ann Beck 7 b. MO
Almarinda Beck 2 b. MO
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1860 United States Federal Census
Name: Moses G Beck
Age in 1860: 29
Birth Year: abt 1831
Birthplace: Kentucky
Home in 1860: Prairie, Montgomery, Missouri
Gender: Male
Post Office: Middletown
Value of real estate: $6000
Household Members: Name Age
Moses G Beck 29 Farmer
Sarah E Beck 24 Wife
Amos A Beck 5 Son
Theodore L Beck 2 Son
Isaac W W Beck 11.12 Son
Coleman Downs 38 Farmer b. VA
Paulina Downs 19 Dau?
Harriett Downs 13 Dau?
Cordelia Downs 7 Dau?
Elizabeth Downs 9.12 Dau?
William Elliott 38 b. NC
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Pike County, Missouri Deaths, 1878-1917
Name: Moses, Capt. Beck
Death Date: Sept. 1862
Issue Date: 2 Mar 1911
Pike County, Missouri Deaths, 1878-1917
Name: Moses, Capt. Beck
Death Date: Sept. 1862
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Missouri Newspaper Death Index
Name: Moses, Capt. Beck
Death Date: 1862
Newspaper: Montgomery City Standard
JR