The ambigous definition of vacuous truth It is no doubt that the vacuous truth is related to material implication  "$P\Rightarrow Q$". We say the material implication statement is true when $P$ is false. However, seems that this is not the definition for vacuously truth. Do we call it vacuously true only when $P$ can never be true? (Seems that all examples are in this way).

More clearly, suppose we have a statement "$P(x)\Rightarrow Q(x)$", and at some domains of $x$ (we may denote the domain by $T_x$), $P(x)$ is true;  at some other domains of $x$ (denote the domain by $F_x$), $P(x)$ is false. Can we say the statement is vacuously true in the domain $F_x$? Is there any real example?

To make the problem more clear, we may check the statement "For all $x$, $P(x)\Rightarrow Q(x)$". Can we say this statement is vacuously true in the domain $F_x$ ($F_x$ is defined as above)?
I would frame the concept as pertaining to first order sentences of the form $$\forall x(P(x)\to Q(x)).$$ Vacous truth is the particular situation when there are no $x$ in the domain satisfying $P(x),$ in which case standard semantics says the statement is true. So yes, whether a first order sentence is vacuously true or not depends on the interpretation, and hence the domain, much as regular truth does. (However this doesn’t usually matter: when we’re talking about math, we generally are making statements relative to some fixed interpretation.)
Do we call it vacuously true only when P can never be true?


No. In classical logic, for any logical propositions P and Q (it doesn't matter if they are true or false), we have:

$P\implies [\neg P \implies Q]$

This is a tautology. See truth table here.