Equivalence of Definitions of Locally Compact Hausdorff Space
$1$ implies $2$
Let $x \in S$.
It is shown that $\mathcal B$ is a neighborhood basis of $x$.
Let $U$ be a neighborhood of $x$.
By Neighborhood in Compact Hausdorff Space Contains Compact Neighborhood, there is a compact neighborhood $V$ of $x$ in $K$ such that $V \subset U \cap K$.
From Intersection of Neighborhoods in Topological Space is Neighborhood we have $W \cap K = V$ is a neighborhood of $x$ in $T$.
From Compact in Subspace is Compact in Topological Space then $V$ is compact in $T$.
Hence $V \in \mathcal B$.
The result follows.
$2$ implies $1$
Follows from Locally Compact Space is Weakly Locally Compact.