I only just noticed your question, Cambrensis, and it's a shame to leave it unanswered. Yes indeed, it is a question of style but also of meaning and emphasis. So your first sentence, I believe, gives a particular emphasis to "in the name of the other farmer" by ending with it.
Tuam quaestionem, Cambrensis, modÃ² animadverti, quae certim responsum meret. ProfectÃ², res suavitatisque et sensÃ»s et emphasis in loquendo est. Ut credo, tuÃ¢ primÃ¢ sententiÃ¢, quae cum clausulÃ¢ "in nomine alterius agricolae" concludit, quaedam emphasis illae clausulae datur.
Also, just as in English, "in the name of -- in nomine" communicates a specific sense when the ablative alone might be ambiguous (as in "by the name of" or "with the name of"). [Similarly, I say "in loquendo" rather than "loquendo" for "in speaking" in my Latin sentence above, 'though I could always be wrong about this, of course.]
Sicut autem anglicÃ¨, dictio "in nomine" minÃ¹s dubia est quam casus ablativus secum.