From some literature I've learned that there are two equivalent definitions of hereditarily collectionwise normal spaces, but struggling with finding a proof.
Definition 1. A topological space $X$ is hereditarily collectionwise normal if every subspace is collectionwise normal (not necessarily Hausdorff).
Definition 2. A topological space $X$ is hereditarily collectionwise normal if for every separated family $\{A_i\}_{i\in I}$ of subsets of $X$ (i.e. a family such that $A_i\cap\operatorname{cl}\bigcup_{j\in I\setminus\{j\}}A_j=\varnothing$ for each $i\in I$), there exists a disjoint family $\{U_i\}_{i\in I}$ of open sets of $X$ such that $A_i\subseteq U_i$ for each $i\in I$.
I've figured out one direction: every discrete family in a subspace $Y$ of $X$ is a separated family in $X$. So Definition 2 implies Definition 1. And I know how to prove both directions for the equivalence of two analogous definitions of completely normal spaces.
Now suppose that $X$ is hereditarily collectionwise normal in the sense of Definition 1, and that $\{A_i\}_{i\in I}$ is a separated family. My plan was to find an open set $Y\subseteq X$ such that $\bigcup_{i\in I}A_i\subseteq Y$ and $\{Y\cap\operatorname{cl}A_i\}_{i\in I}$ is a discrete family in $Y$. Then there would be a disjoint family $\{U_i\}_{i\in I}$ of open sets in $Y$ (and hence in $X$) such that $A_i\subseteq Y\cap\operatorname{cl}A_i\subseteq U_i$. The family $\{A_i\}_{i\in I}$ is discrete in $Y_1=\bigcup_{i\in I}A_i$, but $Y_1$ seems not necessarily open. I also tried $Y_2=\bigcup_{i\in I}\left(X\setminus\operatorname{cl}\bigcup_{j\ne i}A_j\right)$ which is open and contains $\bigcup_{i\in I}A_i$, but I'm not sure if $\{A_i\}_{i\in I}$ is discrete in $Y_2$ (and if so, how to show it). Any help will be appreciated!
Edit: below I list all that I know about discrete families, separated families and collectionwise normal spaces.
Definition 3. Let $X$ be a topological space. A family $\{A_i\}_{i\in I}$ of subsetes of $X$ is called discrete if for every $x\in X$ there is a neighborhood $U$ of $x$ that meets at most one member of the family. The family $\{A_i\}_{i\in I}$ is a separated family if $A_i\cap\operatorname{cl}\bigcup_{j\in I\setminus\{i\}}A_j=\varnothing$ for each $i\in I$.
Proposition 1. A family $\{A_i\}_{i\in I}$ of subsets of a topological space $X$ is discrete if and only if $\{\operatorname{cl}A_i\}_{i\in I}$ is a disjoint family and $\bigcup_{j\in J}\operatorname{cl}A_j$ is a closed set for every $J\subseteq I$.
Proposition 2. A family $\{A_i\}_{i\in I}$ of subsets of a topological space $X$ is separated if and only if the family is a discrete family of $\bigcup_{i\in I}A_i$.
Definition 4. The topological space $X$ is collectionwise normal if for every discrete family $\{A_i\}_{i\in I}$ of closed sets of $X$, there exists a disjoint family of $\{U_i\}_{i\in I}$ open sets of $X$ such that $A_i\subseteq U_i$ for each $i\in I$.
Proposition 3. In Definition 4, one can replace "every discrete family of closed sets" with "every discrete family of subsets", or "a disjoint family of open sets" with "a discrete family of open sets". So we have four equivalent definitions of collectionwise normal spaces.