Generally it is understood that a $T_1$ space is a space in which for two points $x,y$ there exist open sets $U,V$ such that $x \in U, y \notin U, y \in V, x \notin V$. A $T_2$ or Hausdorff space is a space where in addition $U \cap V = \varnothing$.
But for $T_3$ there appear to be at least two seemingly different definitions!
A space is $T_3$ iff it is $T_1$ and regular where regular is used to mean that every neighborhood contains a closed neighborhood.
A space is $T_3$ iff it is $T_0$ and regular where regular is used to mean that given any nonempty closed set $F$ and any point $x$ not in $F$, there exists a neighbourhood $U$ of $x$ and a neighbourhood $V$ of $F$ that are disjoint.
It is not obvious to me that these two defintions are equivalent. It is also not clear to me if the two definitions or regular space are equivalent.
Please could someone explain these definitions to me? I'm a bit confused.