How are properties like "definability" called (in which formulas are involved):
A set $X$ is definable when there is a formula $\phi(x)$ such that $X = \lbrace x : \phi(x)\rbrace$.
It is not a first-order property (not even equivalent to a first-order property!), and it is not a second-order property (but maybe equivalent to a second-order property?)
Is it called a "semantic property" or a "meta(-theoretical) property"?
What is the usual name?
A companion definition of the one above is
A formula $\phi$ is a set-builder when there is a set $X$ such that $X = \lbrace x : \phi(x)\rbrace$.
How are such properties (of formulas) called?
Caveat: I am aware of the fact that treating formulas as sets will lead to problems when definability, truth, or satisfaction comes into play. So I explicitly want to treat them as linguistic entities, different from the objects they are talking about, so no contradictions can arise.