What is the difference between a range image and a depth map?

Your understanding is correct, it's z-values vs distances.

A depth map contains the distances of points in the scene to a plane.

A range image contains the distances of points in the scene to another point. This point is very often the camera's position.

For a depth map the plane that the distances are computed with respect to can be any plane, but it is often parallel to the image plane. Examples of common planes:

• The image plane
• The focal plane
• A plane closer to the object, such that the points closest to the camera have distance zero.

If you of the same scene have:

• A range image with the distances to the camera
• A depth map with the distance to the image plane

these two images will look very similar, and it's not possible to conclude whether it's a range image or depth map, unless you have something you know is a plane in the image.