Learn English – Words for north/south and east/west dimensions

Consider using Transversal or Sagittal. As you pointed out it is hard to describe such a concept because the position you want to describe is extra-relatal from the observer unlike common directional words we use. North and South are close but they are not usually imagined from the first person point of view are are un-conviently pointed to a fixed spot on the horizon in this case. Since you mentioned a third term to coalesce with horizontal and vertical the answer will be a bit subjective because the way the distances and movement between points can be described are similar in both the sagittal and transverse planes. What you are describing is comparing two points along the intersection of the horizontal and vertical planes or more likely between the sagittal and transverse planes.

In this case consider using Transversal. The intersection of the vertical plane with the horizontal plane would form a transverse. This medical definition from thefreedictionary.com describes:

transverse plane of space, n an imaginary plane that cuts the body in two, separating the superior half from the inferior half, and that lies at a right angle from the body's vertical axis.

And the definition for sagital:

sagittal plane

n.

A longitudinal plane that divides the body of a bilaterally symmetrical animal into right and left sections.

Since forward and backward movement at the intersection of the vertical planes is what you are describing, transverse or transversal may be a better choice, however when we describe vertical it is ambiguous whether we are describing a sagittal plane or a coronal

Because of this it is still necessary to describe the relative position between the points. You can use biological or aeronautical terms to describe forward and backward through the transverse plane: dorsal/posterior or aft for back and ventral/anterior and fore for forward.

Edit

After reading some of the other answers it seems clear to me that you are stuck with either a frame of reference to the earth or from the point of view of self. Perhaps the answer here is to abandon the term vertical and horizontal and use for coronal, sagittal, and transversal although that still does not solve your earth point of view in which case you can use latitude, longitude, and altitude.

But keep in mind it's all relative. Pick a system and stay consistent.