What is a 'farmer' in American English?
Solution 1:
My grandfather raised beef cattle (Aberdeen Angus) on an 87-acre farm in central Texas, and I remember asking him once why he didn't refer to himself as a "rancher." He laughed and said that anyone who owned less than a hundred acres of land and called himself a rancher was just kidding himself. In Texas (and I assume elsewhere in the United States) you could qualify for membership in the state's Cattlemen's Association whether you ran a farm or a ranch.
As for the question of who qualified as a farmer, I believe that in Texas the main distinctions were between owning or leasing a farm (which made you a farmer), working on a single farm that someone else owned and operated (which made you a hired hand), and working on many farms at harvest season (which made you a member of, say, a hay-baling crew, if that was the job you did, or a migrant worker, if you picked cotton or fruits or vegetables at multiple farms each harvest season and followed the ripening crops northward as the season progressed).
Solution 2:
I believe the usage varies with community.
For the vast majority of Americans, who live in urban or suburban communities, a farmer is anyone who makes a living by agricultural labor. This would include farm laborers (called farmhands or ranch hands in the US). Ownership/tenancy is irrelevant.
Within rural communities, I believe the UK distinction is made.
ETA - Upon further reflection, I'll modify this to exclude those who raise animals, especially meat cows (oops - beef cattle). Urban/suburban Americans will probably call these folk, both ranchers and ranch hands, cowboys. Those who raise dairy cows are called farmers. For other animals, such as sheep, I doubt that there is a single term which is widely used. Some would call these farmers, some shepherds.
Solution 3:
I think "farmer" is used casually by people, especially those from urban areas, to refer to those in some vague agricultural occupation. If the conversation gets at all specific, however, it will sound odd to refer to a day laborer or migrant worker, etc. as a "farmer". Instead the term will change to one of these or to something like "farmhand". Eg.:
"Didn't you say your brother is a farmer?"
"Well, he's trying to be. He got a job on an organic ranch and he's hoping to work his way up."
Maybe along these lines.