Steve W.,
You have the wrong person. There were two Stephen Booths who enlisted into Rhode Island Regiments.
One who enlisted in Company D 4th Infantry Regiment and the other in Company B 1st Rhode Island Cavalry. Based on the pension index the biographical information you provided is for the R. I. Cav. Stephen Booth - who later moved to Kansas, not the Stephen Booth mentioned by Saragouzou. Apparently, the Kansas Stephen Booth is the uncle of Saragouzou's Stephen Booth.
Here's a link to right Stephen Booth's Find a Grave page:
https://www.findagrave.com/memorial/2250...phen-booth
Here's a rundown of Saragouzou's ancestor's military service
Name: Stephen Booth
Birth Place: Smithfield, Rhode Island
Enlistment Date: 19 Sep 1861
Enlistment Place: Smithfield, Rhode Island
Enlistment Rank: Private
Muster Date: 30 Oct 1861
Muster Place: Rhode Island
Muster Company: D
Muster Regiment: 4th Infantry
Muster Regiment Type: Infantry
Muster Information: Enlisted
Muster Out Date: 15 Oct 1864
Muster Out Place: Providence, Rhode Island
Muster Out Information: Mustered Out
Side of War: Union
Survived War?: Yes
Residence Place: Smithfield, Rhode Island
Title: Register of Rhode Island Volunteers 1861-1865; GAR Dept of Massachusetts 1866-1947 (Sargent)
I found Stephen Booth and his family in the 1841 England and Wales Census, living in Leeds. His father William is listed as having been born in Yorkshire:
Name: Willm Booth
[William Booth]
Age: 35
Estimated Birth Year: abt 1806
Gender: Male
Where born: Yorkshire, England
Civil Parish: Headingley Ward
Hundred: Leeds Borough
County/Island: Yorkshire
Country: England
Street Address:
Occupation: Shop keeper
Registration District: Leeds
Sub-registration District: Kirkstall
Piece: 1344
Book: 2
Folio: 39
Page Number: 9
Household Members:
Name Age
Willm Booth 35
Ana Booth 35
Mary Booth 10
Elizabeth Booth 10
Ann Booth 9
William Booth 7
Henry Booth 4
Stephen Booth 6 Mo
Here he is in the 1850 US census:
Name: William Booth
Gender: Male
Age: 44
Birth Year: abt 1806
Birthplace: England
Home in 1850: Smithfield Districts 1 and 4, Providence, Rhode Island, USA
Occupation: Farmer
Industry: Agriculture
Real Estate: 2000
Line Number: 2
Dwelling Number: 445
Family Number: 509
Household Members & age:
William Booth 44
Nancy Booth 44
Mary Booth 22
Elizabeth Booth 20
Ann Booth 18
William Booth 16
Henry Booth 12
Stephen Booth 8
All household members list their birthplace as England