Where does MSL end and Curiosity begin?

The name of the entire mission is MSL, the Curiosity rover is part of that mission, and the only part operational today. Example of how NASA uses the two names:

Mars Science Laboratory arrived at Mars through technological innovations that tested a completely new landing method. The spacecraft descended on a parachute, then during the final seconds before landing, the landing system fired rockets to allow it to hover while a tether lowered Curiosity to the surface. The rover landed on its wheels, the tether was cut, and the landing system flew off to crash-land a safe distance away.

After separation from the Centaur, MSL consisted of 3 items:

1. Cruise stage
2. EDL system
3. Curiosity.

(cruise stage on left, aeroshell is second and fifth from the left, sky crane third, rover fourth)

The cruise stage:

The cruise stage carried the MSL spacecraft through the void of space and delivered it to Mars. The cruise stage has its own miniature propulsion system, consisting of eight thrusters using hydrazine fuel in two titanium tanks.[117] It also has its own electric power system, consisting of a solar array and battery for providing continuous power. Upon reaching Mars, the spacecraft stopped spinning and a cable cutter separated the cruise stage from the aeroshell.[117] Then the cruise stage was diverted into a separate trajectory into the atmosphere.

The EDL system consists of:

• an aeroshell (heat shield)
• parachute systems
• sky crane

All elements of the cruise stage and EDL system landed on Mars. Curiosity made a soft landing, the other parts crashed in various places:

MRO later made images of the crash site of the cruise stage.