Proof.
The implication (1) \Rightarrow (2) is straightforward and does not need the assumption that X is J-2 or Nagata. Namely, let p \in D and choose an étale neighbourhood (U, u) \to (X, p) such that the pullback of D is a strict normal crossings divisor on U. Then \mathcal{O}_{X, p}^{sh} = \mathcal{O}_{U, u}^{sh} and we see that the trace of D on \mathop{\mathrm{Spec}}(\mathcal{O}_{U, u}^{sh}) is cut out by part of a regular system of parameters as this is already the case in \mathcal{O}_{U, u}.
To prove the implication in the other direction we will use the criterion of Lemma 41.21.6. Observe that formation of the normalization D^\nu \to D commutes with strict henselization, see More on Morphisms, Lemma 37.19.4. If we can show that D^\nu \to D is finite, then we see that D^\nu \to D and the schemes Z_ n satisfy all desired properties because these can all be checked on the level of local rings (but the finiteness of the morphism D^\nu \to D is not something we can check on local rings). We omit the detailed verifications.
If X is Nagata, then D^\nu \to D is finite by Morphisms, Lemma 29.54.11.
Assume X is J-2. Choose a point p \in D. We will show that D^\nu \to D is finite over a neighbourhood of p. By assumption there exists a regular system of parameters f_1, \ldots , f_ d of \mathcal{O}_{X, p}^{sh} and 1 \leq r \leq d such that the trace of D on \mathop{\mathrm{Spec}}(\mathcal{O}_{X, p}^{sh}) is cut out by f_1 \ldots f_ r. Then
D^\nu \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, p}^{sh}) = \coprod \nolimits _{i = 1, \ldots , r} V(f_ i)
Choose an affine étale neighbourhood (U, u) \to (X, p) such that f_ i comes from f_ i \in \mathcal{O}_ U(U). Set D_ i = V(f_ i) \subset U. The strict henselization of \mathcal{O}_{D_ i, u} is \mathcal{O}_{X, p}^{sh}/(f_ i) which is regular. Hence \mathcal{O}_{D_ i, u} is regular (for example by More on Algebra, Lemma 15.45.10). Because X is J-2 the regular locus is open in D_ i. Thus after replacing U by a Zariski open we may assume that D_ i is regular for each i. It follows that
\coprod \nolimits _{i = 1, \ldots , r} D_ i = D^\nu \times _ X U \longrightarrow D \times _ X U
is the normalization morphism and it is clearly finite. In other words, we have found an étale neighbourhood (U, u) of (X, p) such that the base change of D^\nu \to D to this neighbourhood is finite. This implies D^\nu \to D is finite by descent (Descent, Lemma 35.23.23) and the proof is complete.
\square
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