Mauro's answer is very helpful and clear. I'd just like to add to it one point.
Granted the context of the lemma you mention, that lemma isn't just a restatement of soundness but a slight generalization.
Recall that, roughly speaking, soundness says that derivability implies consequence. However, this rough statement is not strictly true for some systems which are intuitively sound. The reason is that derivability may relate formulas with free variables, whereas consequence, as its construed in the notes, applies only to formulas which are true-or-false in a structure, which is to say closed formulas.
As Mauro says, "$\psi$ is a consequence of $\Gamma$" means there's no $M$ such that $M\models \Gamma$ but $M\not\models \psi$. So, and I think this is how it's construed in the cited notes, consequence is a relation on formulas which are true-or-false in an arbitrary structure for the language. For example, $Fx$ will not even be a consequence of $Fx$. Rather, if a formula $\phi$ contains free variables $u_1,\ldots,u_k$, then it is true in $M$ only relative to a substitution $m_1/u_1\ldots,m_k/u_k$ of constants-from-$M$ for those variables.
On the other hand, derivability is a relation on arbitrary formulas. For example, $Fx$ is certainly derivable from $Fx$. Thus, $Fx$ is derivable from itself but not a consequence of itself, and the rough statement of soundness fails.
Happily, consequence can be generalized a notion of free-variable consequence, which applies to open formulas.
$\psi$ is a free-variable consequence of $\phi_1,\ldots,\phi_k$ iff for every $M$ and every $M$-substitution $m_1/u_1,\ldots,m_k/u_k$ for $u_1,\ldots,u_k$ we have that $M\models\psi[m_1/u_1,\ldots,m_k/u_k]$ whenever $M\models \phi_i[m_1/u_1,\ldots,m_k/u_k]$ for all $i$.
Now, the lemma you mention says: "derivability implies free-variable consequence". When "derivability implies consequence" is restricted to closed formulas, it follows immediately from the lemma.
(The need to pass through a special free-variable notion of consequence turns a bit on then notes' choice of a Shoenfield-style rather than Tarskian semantics for the quantifiers.)