what criteria differentiates a rocket from a missile?
A rocket missile is propelled all the way to the target (and may travel more or less horizontally) while a ballistic missile is propelled to top speed on an upward trajectory, and then after the 'burn' is over continues ballistically under gravity (i.e. like a lob in tennis). Ironically, the term 'ballistic missile' has come to mean 'guided', as you point out, although for a purist, 'ballistic' means unguided. However, for most ballistic missiles, the guidance/path-correction forces are small compared with the gravitational forces, so that's reasonable.
A rocket is still under power. A missile is an object that no longer has thrust (e.g. ballistic missiles fall from space, rocks/missiles are hurled from slings, etc.).