type theory notation troubles
$X \vdash Y$ means "The information in $X$ lets you prove that $Y$ is true." This is true in logic in general, as well as in type theory specifically. I often read this as "$X$ says that $Y$ is true."

So,


  If $\Gamma\vdash t\colon T$ and $x\notin\mathrm{dom}(\Gamma)$, then $\Gamma,x\colon S\vdash t\colon T$


translates as


  If we can prove that that $t$ has type $T$ using $\Gamma$, and $\Gamma$ doesn't directly assign a type to $x$, then adding to $\Gamma$ the assumption that $x$ has type $S$ doesn't change the type of $t$,


which is just my way of writing what you wrote.