First we explain how to think about condition (2). The diagonal of an unramified morphism is open (Morphisms, Lemma \ref{morphisms-lemma-diagonal-unramified-morphism}). On the other hand $D^\nu \to D$ is separated, hence the diagonal $D^\nu \to D^\nu \times_D D^\nu$ is closed. Thus $Z_n$ is an open and closed subscheme of $D^\nu \times_D \ldots \times_D D^\nu$. On the other hand, $Z_n \to X$ is unramified as it is the composition $$ Z_n \to D^\nu \times_D \ldots \times_D D^\nu \to \ldots \to D^\nu \times_D D^\nu \to D^\nu \to D \to X $$ and each of the arrows is unramified. Since an unramified morphism is formally unramified (More on Morphisms, Lemma \ref{more-morphisms-lemma-unramified-formally-unramified}) we have a conormal sheaf $\mathcal{C}_n = \mathcal{C}_{Z_n/X}$ of $Z_n \to X$, see More on Morphisms, Definition \ref{more-morphisms-definition-universal-thickening}. \medskip\noindent Formation of normalization commutes with \'etale localization by More on Morphisms, Lemma \ref{more-morphisms-lemma-normalization-and-smooth}. Checking that local rings are regular, or that a morphism is unramified, or that a morphism is a local complete intersection or that a morphism is unramified and has a conormal sheaf which is locally free of a given rank, may be done \'etale locally (see More on Algebra, Lemma \ref{more-algebra-lemma-regular-etale-extension}, Descent, Lemma \ref{descent-lemma-descending-property-unramified}, More on Morphisms, Lemma \ref{more-morphisms-lemma-descending-property-lci} and Descent, Lemma \ref{descent-lemma-finite-locally-free-descends}). \medskip\noindent By the remark of the preceding paragraph and the definition of normal crossings divisor it suffices to prove that a strict normal crossings divisor $D = \bigcup_{i \in I} D_i$ satisfies (2). In this case $D^\nu = \coprod D_i$ and $D^\nu \to D$ is unramified (being unramified is local on the source and $D_i \to D$ is a closed immersion which is unramified). Similarly, $Z_1 = D^\nu \to X$ is a local complete intersection morphism because we may check this locally on the source and each morphism $D_i \to X$ is a regular immersion as it is the inclusion of a Cartier divisor (see Lemma \ref{lemma-strict-normal-crossings} and More on Morphisms, Lemma \ref{more-morphisms-lemma-regular-immersion-lci}). Since an effective Cartier divisor has an invertible conormal sheaf, we conclude that the requirement on the conormal sheaf is satisfied. Similarly, the scheme $Z_n$ for $n \geq 2$ is the disjoint union of the schemes $D_J = \bigcap_{j \in J} D_j$ where $J \subset I$ runs over the subsets of order $n$. Since $D_J \to X$ is a regular immersion of codimension $n$ (by the definition of strict normal crossings and the fact that we may check this on stalks by Divisors, Lemma \ref{divisors-lemma-Noetherian-scheme-regular-ideal}) it follows in the same manner that $Z_n \to X$ has the required properties. Some details omitted. \medskip\noindent Assume (2). Let $p \in D$. Since $D^\nu \to D$ is unramified, it is finite (by Morphisms, Lemma \ref{morphisms-lemma-finite-integral}). Hence $D^\nu \to X$ is finite unramified. By Lemma \ref{lemma-finite-unramified-etale-local} and \'etale localization (permissible by the discussion in the second paragraph and the definition of normal crossings divisors) we reduce to the case where $D^\nu = \coprod_{i \in I} D_i$ with $I$ finite and $D_i \to U$ a closed immersion. After shrinking $X$ if necessary, we may assume $p \in D_i$ for all $i \in I$. The condition that $Z_1 =  D^\nu \to X$ is an unramified local complete intersection morphism with conormal sheaf locally free of rank $1$ implies that $D_i \subset X$ is an effective Cartier divisor, see More on Morphisms, Lemma \ref{more-morphisms-lemma-lci} and Divisors, Lemma \ref{divisors-lemma-regular-immersion-noetherian}. To finish the proof we may assume $X = \Spec(A)$ is affine and $D_i = V(f_i)$ with $f_i \in A$ a nonzerodivisor. If $I = \{1, \ldots, r\}$, then $p \in Z_r = V(f_1, \ldots, f_r)$. The same reference as above implies that $(f_1, \ldots, f_r)$ is a Koszul regular ideal in $A$. Since the conormal sheaf has rank $r$, we see that $f_1, \ldots, f_r$ is a minimal set of generators of the ideal defining $Z_r$ in $\mathcal{O}_{X, p}$. This implies that $f_1, \ldots, f_r$ is a regular sequence in $\mathcal{O}_{X, p}$ such that $\mathcal{O}_{X, p}/(f_1, \ldots, f_r)$ is regular. Thus we conclude by Algebra, Lemma \ref{algebra-lemma-regular-mod-x} that $f_1, \ldots, f_r$ can be extended to a regular system of parameters in $\mathcal{O}_{X, p}$ and this finishes the proof.