I used to think that there was a point at which you escaped the Earth's gravitational pull. This is technically false, since every object in the universe is attracted towards every other object. Is there a term for the point at which we consider it negligible? I think of the Oort Cloud and it's boundaries around the Sun's significant gravitational pull
Although related to similar concepts, the terms sphere of influence and Hill sphere are used in different contexts, as already pointed out by someone answering the following question on the astronomy branch.
Basically the Hill sphere is related to the stability of an object (no. 2) orbiting another one (no. 1), subjected to the perturbations of a third body (no. 3). If the orbit of (2) lies outside the Hill sphere of (1), then the gravitational perturbations induced by (3) will eventually pull out (2) from its orbit about (1). Otherwise the orbit of (2) is said to be stable.
On the other hand the sphere of influence is mainly used in the context of the patched conic approximation, a method used to simplify the trajectory computations of a spacecraft in a complex multiple-body gravitational field. The idea is to discretize the space in different zones in which one can consider the spacecraft to be subjected only to the gravitational field of a single object, neglecting the other bodies. These zones are the spheres of influence.
Given this distinction, I would say that the answer to the original question is sphere of influence and yes, it false that at a precise point in space a probe escapes the Earth's gravity field, but it is an acceptable simplification.
I'd like also to underline a concept that often seems to be taken lightly. Both the Hill sphere and the sphere of influence of a body in space (usually a planet) can only be defined in relation with another celestial body (usually the Sun).
Hope this could be useful.
You could say you are outside the Hill sphere of another object. From Wikipedia:
An astronomical body's Hill sphere is the region in which it dominates the attraction of satellites. [...] One simple view of the extent of the Solar System is the Hill sphere of the Sun with respect to local stars and the galactic nucleus.