「ロバ文」の説明は以下のようなものです。（http://www.reddit.com/r/linguistics/comments/28tzau/why_is_this_firstorderization_of_the_donkey/）
The problem with donkey sentences is not that they cannot be translated to FOL, but rather that the translation is inconsistent with other data.
For example:
John owns a donkey.
∃y[Dy ∧ Oyj]
Every farmer owns a donkey.
∀x[Fx -> ∃y[Dy ∧ Oyx]]
The above sentences indicate that the indefinite a corresponds to the existential quantifier ∃. Moreover, there's no disjunction going on. This would be incorrect:
John owns a donkey.
∃y[Dy ∧ Oyj] ∨ ~∃y[Dy ∧ Oyj] <-- tautology
Every farmer owns a donkey.
∀x[Fx -> (∃y[Dy ∧ Oyx] ∨ ~∃y[Dy ∧ Oyx])] <-- tautology
Let's now put owns a donkey into a relative clause and add a different verbal predicate, with no anaphoric relation.
Every farmer who owns a donkey is rich.
∀x[(Fx ∧ ∃y[Dy ∧ Oyx]) -> Rx]
Once again, a is translated as a single existential quantifier. The following would be incorrect:
Every farmer who owns a donkey is rich.
∀x[(Fx ∧ (∃y[Dy ∧ Oyx] ∨ ~∃y[Dy ∧ Oyx])) -> Rx]
=∀x[Fx -> Rx]
(Every farmer is rich.)
Turning now finally to the donkey sentence, i.e. changing is rich to beats it in the sentence above, the foregoing data leads us to this translation:
Every farmer who owns a donkey beats it.
∀x[(Fx ∧ ∃y[Dy ∧ Oyx]) -> Byx]
As you probably know, the problem with this formula is that y is free in Byx.
Summarizing, you're right that the problem with your formula is the disjunction: there's no consistent way to translate a that would allow it to be both a normal existential and a disjunctive existential, so to speak.
There's also another problem with your formula: the part corresponding to the relative clause who owns a donkey comes after the conditional arrow, whereas in a sentence like Every farmer who owns a donkey is rich, it comes before (see above).
Your formula is rather the translation of a sentence like: Every farmer owns and beats a donkey or does not own a donkey.
以上のように専門用語なしで説明することは出来ない代物です。
結論：自然言語を形式論理式で正確に表記することは難しい。人間は状況を理解して柔軟に解釈できる頭脳を持っているが、コンピュータは形式通りにしか理解できないので「all」と「every」のニュアンスの違いを区別できない。
以上、参考になれば幸いです。
補足
この辺の「難しさ」は「各農夫が一匹ずつロバを飼っている。各農夫が、それぞれが飼っているロバを叩く」という解釈と「農夫みんなで一匹ロバを飼っている。各農夫がその一匹のロバを叩く」というような解釈・・・この二つの解釈の存在に起因するような性質のものですか？まったくトンチンカンな質問だったらごめんなさい。