Note that there is no absolute definition of "at rest", it will depend on your frame of reference. In the frame of the combined object after an inelastic collision, it is at rest. In any other reference frame, it is not. Whether the combined object is at rest or not will depend entirely on your point of view.
In a perfectly inelastic collision, the bodies stick together and move with zero relative velocity. In any reference frame, the two bodies have the same velocity, and in the frame of the combined body, that velocity is zero. Any other situation that has a non-zero relative velocity between the bodies is not a perfectly inelastic collision.
In a non-perfect inelastic collision, there is no reference frame where both objects have zero velocity and are "at rest". No matter what reference frame you pick, at least one of the objects will be moving. So in a way, your proposed definition of inelastic collisions actually just reiterates what an inelastic collision is - it is the only type of collision where both bodies come to rest in some particular reference frame. Imperfect inelastic collisions do not have both bodies come to rest in any reference frame.
In the reference frame of the combined body, it is at rest, so its kinetic energy is 0. Before the collision, at least one of the bodies was moving, so there was non-zero kinetic energy. The maximum possible kinetic energy loss has taken place in the inelastic collision - in the frame of the combined object, all of the kinetic energy has been dissipated. In other reference frames, the combined object will still have kinetic energy, but it's still the maximum possible loss.