This is my best understanding:

Runners are a line (flat-tack, cord, or Dyneema, or w/e) that 'run' from one thing to another, usually some rock protection to our rope. They can be in a "rabbit runner" form (two little loops with a line between), or in a "sling" form (a continuous loop), or in a un-looped line, ready to be constructed into a runner.

A sling is a subset of a kind of runner, and is a loop. It can be a several sizes. It can act as a runner, between two points like rock protection and a rope, or can be slung around a fixed point (like a rock, flake, or prominence) to create a protected point itself. An alpine draw is a form of runner in the shape of a sling.