There's two statements:
(i) ∃x (PresentKingFrance(x) → Bald(x)); (ii) ∃x (PresentKingFrance(x)) → ∃x (PresentKingFrance(x) ∧ Bald(x)).
I'm not an expert in FOL, but I still understand a little bit. A little bit, but not this.
What's the difference between these two statements? Can you explain this to me with details by the means of natural language? Or maybe even graphically....
I also don't understand why can't we transform (ii) into ∃x(PresentKingFrance(x) → (PresentKingFrance(x) ∧ Bald(x)))
If two formulas seem confusingly similar like $i$ and $ii$, then you can always take their negations and compare to see if they’re actually logically equivalent.
In this case, the negation of $i$ is saying everyone is a non-bald present King of France, while the negation of $ii$ is saying that there is a present King of France, and whoever is the present King of France is not bald.
These are clearly different, since the negation of $ii$ is not saying that everyone is a present King of France, while the negation of $i$ is. This is reflected in the non-negated versions, namely since $i$ is satisfied if just one person is not the present King of France, while $ii$ is not necessarily satisfied under those same conditions.
Also, note that the formula you mention in your final paragraph is indeed equivalent to $i$ since
$A \to B$
and
$A \to (A \land B)$
are logically equivalent.