First, I think the usual notation for disjunction is $\vee$. $\wedge$ usually means "and".
Also, the term "or" as in $A \vee B$ is defined as follows :
A is True, B is True, then $A \vee B$ is True
A is True, B is False, then $A \vee B$ is True
A is False, B is True, then $A \vee B$ is True
A is False, B is False, then $A \vee B$ is False.
Therefore the term "or" (in this context) already includes the case when both are true. So for your expression $(\exists x)(P(x) \vee Q(x))$, it suffices to translation as follows :
There exists an $x$ such that $P(x)$ or $Q(x)$.
In this context, most people will understand the intended meaning of the "or".