The question of what a "farm" seems simple. It's actually complex. From a bureaucratic standpoint it depends on the purpose of the farm program.
Back in the day, Farmers' Home Administration would not talk of a farm, but a farming operation, which as I understand it included all the land, animals and equipment belonging to or operated by the "farmer". Essentially when FmHA made a loan to a "farmer", they wanted to consider everything which could impact the viability of the loan. They didn't care about location.
Soil Conservation Service cared only about location. They worked with the conservation practices on a plot of land, their offices served soil and water conservation districts (usually but not always a county) so what a farmer did in county B was irrelevant to conservation in county A.
ASCS was ambivalent, having to deal with both people and land, both landowners and operators/producers. In the days when disaster programs were uppermost, we wanted to combine land to spread losses and production over the widest area. In the days when production adjustment was foremost, we wanted to divide land, so the operator had the least ability to designate less-productive land as her set-aside/conservation acreage. When programs shifted (as in the early 80's, our rules were often out-of-date.
Apparently today's programs may have impacted FSA's rules on farm constitution. DTN has pieces from Marcia Zarley Taylor and Chris Clayton on the issue. Because some payments under the new farm bill are now determined using county-level data, whether land located in more than one county is administratively consider to be one farm and located in one county can make a difference. The articles point out the possibility of losses (farm is located in county B when county A has a higher payment rate). As usual, they don't point out the possibility of what one might call "windfalls", the farm is located in county A even though much of the land is in the lower rate county B.