How is the gravity of a massive objects like earth and sun defined

How is it defined in terms of "where" the gravity is "located?

Is it just arbitrary to say it's located at it's center according to Newtonian physics?

How does it change with Einstein's general relativity? Is it still defined as coming from a center point?

If it is defined as the center of an object, is that not just an arbitrary assumption?

Each atom of any body has a gravitational field as described by Newton. The aggregate field of all the atoms behaves as though the entire mass is located at a single point, which we call the center of mass.
Nothing arbitrary at all about it.