Separated sets in Basic Topology by Armstrong

Question

In Armstrong's Basic Topology, page 58, subsets $A$ and $B$ of a space $X$ are separated if

$\bar{A} \cap \bar{B} = \emptyset$.

However, anywhere else I have looked defines separated sets with the condition

$A \cap \bar{B} = \bar{A} \cap B = \emptyset$.

These are clearly not equivalent, so is the definition in Armstrong non-standard?

Edit: removed faulty example