For modular curves over schemes there are two main references that I use, namely Deligne Rapoport [DR], and Katz-Mazur [KM]. However I recently noticed that there is a difference in conventions in these two articles.
Let $E$ be an elliptic curve over a scheme $S$.
In [DR, page 68] a level $N$ structure is defined as an isomorphism $\alpha: E[N] \to (\mathbb Z/N\mathbb Z)^2$. Note that in [DR] the integer $N$ is assumed to be invertible on $S$.
While in [KM, page 68], $N$ is not assumed to be invertible, and a level structure is defined as a homomorphism $\phi: (\mathbb Z/N\mathbb Z)^2 \to E[N]$ satisfying some extra conditions. These extra conditions are weaker then being an isomorphism, but are equivalent to being an isomorphism if $N$ is invertible on $S$.
Main question: If $N$ is invertible on $S$ is there a mathematical reason why one might want to use the [DR] convention of having the map go from $E[N]$ to $(\mathbb Z/N\mathbb Z)^2$ instead of from $(\mathbb Z/N\mathbb Z)^2$ to $E[N]$ as in [KM]?
Note that I do already have an answer the converse of the main question. As there is a reason to prefer the [KM] convention of having $(\mathbb Z/N\mathbb Z)^2$ as the source and $E[N]$ as the target. Indeed when $N$ is not invertible on $S$ then there are cases where the $\phi$ in [KM] doesn't have an inverse. And one of the novelties in [KM] over [DR] is allowing certain $\phi$ that do not have an inverse in order to get a moduli space over $\mathbb Z$ that has certain desirable properties, like being proper.
[DR] Deligne, Pierre; Rapoport, M., Moduli schemes of elliptic curves, Modular Functions of one Variable II, Proc. Int. Summer School, Univ. Antwerp 1972, Lect. Notes Math. 349, 143-316 (1973). ZBL0281.14010.
[KM] Katz, Nicholas M.; Mazur, Barry, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, 108. Princeton, New Jersey: Princeton University Press. XIV, 514 p. hbk: {$} 78.00; pbk: {$} 22.50 (1985). ZBL0576.14026.
