Definition(1)
A space $X$ is $T_3$ iff 'For any closed set $F$ and a point not in $F$, there exist non overlapping open neighborhoods.
Definition(2)
A space $X$ is $T_3$ iff 'For any nonempty closed set $F$ and a point not in $F$, there exist non overlapping open neighborhoods.
First of all, 'wikipedia' refers to the second definition. However, 'proofwiki' refers to the first definition. And i don't understand why wikipedia specifically restricted the definition to nonempty closed sets.
Which one is widely accepted?
Moreover, what's a regular space? Many texts define it different from one another. Is it just $T_3$ generally?
Both the definitions are equivalent. Since the condition for empty set is always true, so it redundant to mention that nonempty closed sets.
According to Kelly, what ever definition you have given is the definition of a regular space and a regular $T_1$ space is a $T_3$ space.