Define the star $St(S, \mathcal{U})$ of a set $S$ with respect to the collection of sets $\mathcal{U}$ as $St(S, \mathcal{U}) := \bigcup \{U \in \mathcal{U} \mid S \cap U \neq \varnothing\}$. If $S=\{x\}$, we write $St(x,\mathcal{U})$.
Steen and Seebach define a star refinement of a cover $\mathcal{U}=\{U_\alpha\mid\alpha\in A\}$ of a set $X$ as a cover $\mathcal{V} = \{V_\beta\mid\beta\in B\}$ such that $\{St(x,\mathcal{V})\mid x \in X\}$ is a refinement of $\mathcal{U}$ (basically).
Dugundji defines a star refinement of a cover $\mathcal{U}=\{U_\alpha\mid\alpha\in A\}$ of a set $X$ as a cover $\mathcal{V} = \{V_\beta\mid\beta\in B\}$ such that $\{St(V_\beta,\mathcal{V})\mid \beta \in B\}$ is a refinement of $\mathcal{U}$.
I'm wondering if these are equivalent, and if so, are the collections $\mathcal{U}$ and $\mathcal{V}$ are required to be covers?
A Dugundji star refinement (for lack of a better term) is also a Steen and Seebach star refinement even if we drop the requirement that $\mathcal{U}$ and $\mathcal{V}$ are covers (assuming $\mathcal{U}$ is nonempty): Suppose $x \in X$. Either $x \in \bigcup\mathcal{V}$ or $x \notin \bigcup\mathcal{V}$. If the former, then $x \in V_\beta$ for some $\beta \in B$ and we have $St(x,\mathcal{V}) \subseteq St(V_\beta,\mathcal{V}) \subseteq U_\alpha$ for some $\alpha \in A$ since any $V_\gamma \in \mathcal{V}$ containing $x$ must intersect $V_\beta$. If the latter, then $St(x,\mathcal{V})=\varnothing \subseteq U_\alpha$ for any $\alpha \in A$.
I feel like I must be missing something simple, because I can't seem to prove the converse (with or without the covering assumption).
Both types of refinements are (implicitly or ecplicitly) only applied to covers.
The type with stars of points is properly called a barycentric refinement.
So $\mathcal{U} \prec_b \mathcal{V}$ means that both are covers of the space $X$ in question (often open, but with paracompactness etc. one also considers closed covers as well) and for every $x \in X$, $\operatorname{St}(x, \mathcal{U}) \subset V$ for some $V \in \mathcal{V}$.
The other type is a real star refinement:
$\mathcal{U} \prec_\ast \mathcal{V}$ iff $\mathcal{U}$ and$\mathcal{V}$ are covers of $X$ and $\forall U \in \mathcal{U}$ there exists some $V \in \mathcal{V}: \operatorname{st}(U,\mathcal{U}) \subseteq V$
See wikipedia for this as well. Note that $\mathcal{U} \prec_\ast \mathcal{V}$ implies $\mathcal{U} \prec_b \mathcal{V}$ if we have covers. So one notion is stronger than the other for covers (if points are uncovered the star of such a point is empty)
A basic fact: if we have 3 covers $\mathcal{U}, \mathcal{V}, \mathcal{W}$ of $X$ then $\mathcal{U} \prec_b \mathcal{V}$ and $\mathcal{V} \prec_b \mathcal{W}$ implies $\mathcal{U} \prec_\ast \mathcal{W}$.
(this is a nice exercise, try it). This means that if we assume that every open cover has a barycentric refinement, we get also that every open cover has a star-refinement: just take two successive barycentric ones. Hence they're both used in some definitions, but he definitions are not equivalent. There are barycentric refinements that are not star-refinements.