Can a distinction between intrinsic and extrinsic properties of general sets
a) be defined rigorously and
b) be used fruitfully? (References?)
An intrinsic property of a set $M$ is supposed to be about (the) elements contained in $M$, an extrinsic property is supposed to be about (the) sets containing $M$.
Prototypical examples of intrinsic properties $P$ of a set $X$ are given by formulas of the form
$$P(X) :\equiv \forall x\ x\in X\Rightarrow \phi(x)$$
or
$$P(X) :\equiv \exists x\ x\in X\wedge \phi(x)$$
Prototypical examples of extrinsic properties $P$ of a set $x$ are given by formulas of the form
$$P(x) :\equiv \forall X\ \phi(X) \Rightarrow x \in X$$
or
$$P(x) :\equiv \exists X\ \phi(X) \wedge x \in X$$
(How) can these examples be subsumed under a general definition?
Side note: We find the intrinsic/extrinsic dichotomy in differential geometry. Is it defined rigorously at least here? And in which "specialized" fields else - other than set theory?