Both records cited above indicate that Private William Townsley and others in Capt. Kain's Company served from August 24th to September 24th, 1812.Service of one month suggests that Kain's Company was a militia company.Militia service of thirty days was obligatory for every male above a certain age.
www.ancestry.com "War of 1812" records show:
Name: William Townsley COMPANY: CAPT. KAIN'S MTD. COMPANY, OHIO MILITIA. Rank - Induction: PRIVATE Rank - Discharge: PRIVATE Roll Box: 210 Roll Exct: 602
If you wanted to see exactly what is recorded in that record, you would have to find an LDS Family History Center, or a Genealogical Society, or a Library that holds the microfilm containing that record.It is my understanding that said record will consist of "a compiled military service record consists of a card or cards on which is recorded information about a solider that was collected from muster rolls, returns, pay vouchers, hospital registers, prison records, and other records. The abstracted information may include references to wounds, hospitalization, absence from the unit, court martials, and death. This will not represent all of the information about a soldier scattered among a variety of records." (See https://eservices.archives.gov/orderonline/start.swe?SWECmd=GotoView&_sn=1uPlFwiH7A8mMVGSkeXebu8Oo0yE0gzgxCbv3AJpxTU_&SWEView=GPEA+Find+Request+View&SWEHo=eservices.archives.gov&SWETS=1213469117&SWEScreen=GPEA+Find+Request+Screen )