I'm afraid that I can't disentangle what that FamilySearch (FS) reference actually means. According to the index on the URL you gave, the entry comes from the "Great Britain, War Office Registers, 1772-1935", and according to the page on that collection, "The records are held at The National Archives in Kew under WO 25". WO 25 is, I'm afraid, a bundle of various types of register books, so I can't tell anything more about the record that FS have indexed.
If I go to FindMyPast (FMP) and try various searches on their Military stuff, looking for James Vincent, b 1779 +/- 5y, it's surprising how many matches there are. There are at least three born in Wilton, Wilts, though I'm not convinced they all qualify by year of birth.
Perhaps the most interesting one is this, as it roughly matches by age to your baptism:
First name(s) James
Last name Vincent
Discharge corps 3rd Garrison Battalion
Document type Discharge
Archive reference WO 97
Birth parish Wilton
Series Wo 97 - Chelsea Pensioners British Army Service Records 1760-1913
Birth town Salisbury
Birth county Wiltshire
Rank: Serjeant Major
Record set British Army Service Records
Regiment Royal Garrison Battalions
Subcategory Regimental & Service Records
This is a document discharging him in Feb 1815 at Malta from that regiment. It gives a list of the regiments that he served in - which starts in 1795 in the Royal Marines before moving over the the British Army proper - and his age (36, i.e. b abt 1779).
The above may also be a/v on Fold3 as Ancestry has an index to a record with similar details (assuming we match "Wilton" on FMP to "Willow" on Ancestry).
I suspect that the above is in the TNA Catalogue as:
Reference: WO 97/1145/399
Description: JAMES VINCENT
Born WILTON, Wiltshire
Served in 66th Foot Regiment; 3rd Garrison Battalion; 2nd Garrison Battalion; Royal Marines
Discharged aged 36
I have no knowledge of whether anything useful survives in Royal Marine records for the Serjeant Major (note archaic spelling!). Army documents of this period are (Muster Books and Paylists aside) mostly related to pensions - even what looks like an Attestation document (created on joining the Army) is usually a copy created for pensions purposes.
The most detailed documents tend to be the discharge documents as per the WO 97 example mentioned above. There can also be other pages created solely for the purposes of administering pension payments. In my personal view (i.e. I'm guessing) that's probably what the FamilySearch document is. Unless I saw the original I'm not sure I'd even believe the WO 25 reference and I certainly don't believe "Draft Registration" - that sounds like FamilySearch creating a list of events and being unwilling to update it.
Your tasks could be to
- get access to that FamilySearch image;
- check the image for the Serjeant Major to see if it can be linked to your chap and / or to the FS image.
Note that if it is the Serjeant Major, the list of regiments on his WO 97 Discharge Document is all you're going to get, apart from Muster Books & Paylists. These show quarterly snapshots of everyone in a regiment / battalion / depot but can often, on the first and last such entry in a unit, indicate previous or next service. Aside from a very few on Ancestry, these are all only available at TNA, Kew, in the UK.