Farmer is profession.
For me the word: peasant is more like mental state being. For me it means ignorance, stupidity, mental poverty, simple-mindedness, naivity, following backwards social customs slavishly, the lack of independent thinking.
Person could be really wealthy but still have "the mind of peasant".
Pleb in UK slang (coming from Latin world for common people "plebeii") means the same aka stupid person... person of inferior intelligence.