Nitpick: you're thinking of a ballistic missile. Cruise missiles are more like exploding fixed-wing drones. The V-1 flying bomb was a cruise missile, and the V-2 rocket was a ballistic missile.
If a fully loaded (cryogenic fuel + oxidizer) SpaceX rocket did a belly flop into downtown Orlando I suspect this distinction would be quite meaningless to the locals!
Not sure what exactly are you nitpicking here - ballistic missiles have propulsion for a short time after lunch and afterwards continuing on a ballistic (hence the name) trajectory towards the targets. Cruise missiles have propulsion usually during most of the flight.
A rocket straying off course could either smash into city under full power (thus being a cruise missile) or have its propulsion cut off remotely beforehand (thus being a ballistic missile), depending on a scenario. What kind of a correction were you making?
Cruise missiles are designed to rely on aerodynamic lift (and typically use air-breathing engines, although there have been some "boost-glide" weapons that used rocket engines).
