Undoubtedly, these terms play essential roles in (pure) mathematics. My problem is that I have feelings what they mean in different fields, such as, differential geometry (abstract manifolds vs. embedded ones), algebraic geometry (more down to earth, the study of Riemann surfaces and algebraic curves) when our objects can be embedded differently and each embedding gives us (I think) extrinsic properties rather than intrinsic ones, and intrinsic properties do not have anything to do with embeddings.

What I would like to know are as follows;

Can they be defined, precisely?

How can one recognize which properties are coming from intrinsic properties and which are not?

I would appreciate any comments in helping me understand them better.