I think you can use all those terms.
According to Reta Vortaro, the root word is farmi, which means leasing land to grow crops. So, in theory, if you own the property you use to grow crops, you are a bienisto. If you lease the property, you're a farmisto. To make it more fun, you can say that the land the farmisto works on is a farmo or a bieno.
However, in one of the remarks (for farmo), it notes that the idea of leasing can disappear and it refer only to an agricultural enterprise.
Checking on Lernu shows that besides bienisto and farmisto, you can call a farmer terkulturisto, terkultivisto, agrikulturisto, kampulo, or agrokulturisto....there's a lot of farmers out there!
Lernu.net lists "bieno" as "estate/property/farm/ranch". And "bienulo (bien·ul·o ← bien·o) landowner", but "biendomo (bien·dom·o ← dom·o) farmhouse".
I would think that if a farmer is a "farmisto" (which Duolingo has already taught us), then "bieno" being a farm doesn't make any sense, or at least seems inconsistent. So maybe "bieno" just doesn't translate well to English? Maybe I'm just thinking like an American again?
For those interested to know, there is a good discussion about this in the DB Duolingo Group: https://www.facebook.com/groups/duolingo.esperanto.learners/search/?query=bieno
I remember an earlier tips and notes section said to use the "-e" ending after "esti" but why does it sometimes end in "-a"?