a heightmap is three-dimensional, where the third dimension is usually represented with color or contour lines.

A mathematically ideal sheet of paper in a 3d-dimensional space can be addressed with both 2D coordinates, within the reference frame of the paper, and 3D coordinates, within the 3D space, but given that the paper has a location in space, 2D coordinates are sufficient to specify a point in 3D space.

You can do the same with geography. It's why generally only specify geographical coordinates with latitude and longitude, altitude is a given from those two, since you're as unlikely to be hovering in the air as you are to be immersed in bedrock.