An argument is valid iff :
there is no possible situation in which the premisses would be true and the conclusion false.
An argument is valid because it shows a "pattern", like :
All $A$ are $B$
$s$ is $A$
Therefore, $s$ is $B$.
We can think to it as a "form" to be filled : filling the "form" in the correct way, we get an instance of that "argument-pattern", like :
"All $Men$ are $Mammals$
$socrates$ is a $Man$
Therefore, $socrates$ is a $Mammal$",
with the feature that : if all the premisses are true, then also the conclusion is.
We can think at this beautiful feature as a way to "transfer" truth from premises to conclusion : we cannot "transfer" it if it is not present in the premisses.
What happens if we "fill the form" correctly, but with false premisses ? The "argument-pattern" does not change, but the antecedent of the definition above : "if all the premisses are true, ..." is not satisfied, and thus it gives us no clue about the truth or falsity of the conclusion.
This is what happens with :
"All $Cups$ are $Green$
$socrates$ is a $Cup$
Therefore, $socrates$ is $Green$."
Both premises are false, and thus the falsity of the conclusion does not invalidate the "argument-pattern".
We can have also cases more tricky than that; consider :
"All $Cows$ are $Mammals$
$socrates$ is a $Cow$
Therefore, $socrates$ is a $Mammals$."
In this case the conclusion is true, but not all the premises are. The argument is still valid, but we have applied it in the "wrong way" and thus we are not licensed to assert the conclusion on the ground of the argument alone.
In other terms, if we want to assert the (true) fact that "$socrates$ is a $Mammal$", we can do it e.g. according to some empirical knowledge, but not on the ground of the above (valid) argument, because the corresponding "argument-pattern" licenses us to "flag" the conclusion as true only when we have inferred it from true premisses.
Regarding the question :
what does the "truth" or "falsity" of a statement refer to? The "general truth" of the statement, or some other type of "truth" and "falsity"?
there are no "multiple" meaning of true in mathematics and logic (until we do not try to elucidate the concept of truth at a more philosophical level).
The notion of truth (in the form of the predicate "is true") applied to mathematical statements amounts to saying that a statement is true if and only if things are as they are said to be in the statement; more formally :
for any mathematical statement $\phi$, "$\phi$ is a true mathematical
statement" is equivalent to $\phi$ itself.
Applied to our discussion above, "$socrates$ is a $Mammal$" is true because $socrates$ is a $Mammal$, while "$socrates$ is $Green$" is false because $socrates$ is not $Green$.