No Gold is Silver.
Let $G(x) :$ $x$ is Gold
$S(x) :$ $x$ is Silver
Interpretation 1 : For all $x$, If $x$ is Gold then it is Not silver.
$\forall x(G(x) \rightarrow \sim S(x))$
Interpretation 2 : There does not exist any God which is Silver. Or It is Not the case that there exists some $x$ which is Gold and Silver.
$\sim \exists x(G(x) \wedge S(x))$
Your Second Formula :
$ \sim \forall x(G(x) \rightarrow S(x))$ : It is Not the case that All the Gold are Silver. Or There exists some $x$ which is Gold but Not Silver.
So, This is Not a Valid formula for "No Gold is Silver".