Is there any difference between
$(\forall e) ((\exists N=N(e)) (P(N,e)))$.
and
$(\forall e) ((\exists N)(P(N,e)))$
Should we read this two statements differently? What does this $N=N(e)$ stand for? To me it is clear that $N$ depends somehow on $e$ from both statements.
On
Trying to convey in more or less conventional words your formulas, we can say the first one says "For all $\,e\,$ there exists a function $\,N\,$ of $\,e\,$ s.t. $\,P(N,e)\,$ , whereas the second one omits the words "a function" above.
We're assuming, of course, that everybody knows what kind of beings $\,N,e ,P\,$ etc. are...
The second formulation is entirely rigorous. The first formulation aims at stressing the fact that N may depend on e, and while doing this sacrifices rigor. This is useless since the fact that N may depend on e is indicated nonambiguously by the order of the quantificators.
One can advise to stick to the second form and to add afterwards a formulation similar to: Let N(e) denote some N such that P(N,e) holds.