Star refinement and paracompactness

Question

If $\mathcal{U}$ is a cover of a space $X$ and $S\subseteq X$, define the star of $S$ with respect to $\mathcal{U}$ as $\text{st}(S,\mathcal{U})=\bigcup\{U\in\mathcal{U}\mid S\cap U\neq\emptyset\}$.

1. According to wikipedia, a Hausdorff space is paracompact iff it is fully normal; it is also defined there that a space is fully normal if every open cover has an open star refinement. In the same page it says a cover $\mathcal{U}$ is a star refinement of $\mathcal{V}$ if for any $x\in X$, $\text{st}(\{x\},\mathcal{U})\subseteq V$ for some $V\in\mathcal{V}$. However the wikipage for star refinement says it should be for any $U\in\mathcal{U}$, $\text{st}(U,\mathcal{U})\subseteq V$ for some $V\in\mathcal{V}$, which is a priori stronger. Do these give the same definition? If not which one is the correct definition of fully normal spaces?

2. Consider the following property: any open cover $\mathcal{V}$ has an open refinement $\mathcal{U}$ such that any $U\in\mathcal{U}$ intersects only finitely many other members of $\mathcal{U}$. This property easily implies paracompactness. Is the converse true (maybe under mild assumption)? If not does this property has a name?

Answer 1

By fact 1 in this note it follows that the notion of every cover having a star-refinement (with the $U$ as first component) is equivalent to having a barycentric open refinement for each open cover (with $\{x\}$ as first component).

So we can use either one for the definition of fully normal.

A cover such that every element intersects only finitely many other elements is calls star-finite and the property that every open cover has a star-finite open refinement is called strongly paracompact and indeed implies paracompact (but not always conversely). Asking for a point-finite refinement instead (every point is in finitely many members) is called weakly paracompact (or metacompact in some texts) and is strictly weaker than paracompactness. E.g. A metric space is always paracompact but only strongly paracompact when we can embed it into $[0,1]^\omega \times D^\omega$ for some discrete space $D$ (Morita's theorem).

All of this is proved in Engelking's book on General Topology (1989 edition, Ch. 5), if you need a reference.