(I am skipping any setup stuff and speaking roughly)
One fact that I am sure of is that a definable subset $X$ is fixed by all automorphisms of the (super)structure.
I simply wonder the converse:
"If a subset $X$ is fixed by all automorphisms, then $X$ is a definable set."
Is this true (in general)? If not, could you provide some counterexample and the cases/conditions that the latter assertion is true?