Just some thoughts

• The need to name these positions.
• You are confident these positions need to be named. Do they actually? See next.
• Think outside the box tetrahedron
• there are three coordinates which relate to a point in a system. This does not necessitate that a 3D model inside the system would be a cube. Would you be concerned about naming all the sides of an octahedron. Maybe there is scope for a tetrahedral model to navigate such a system.
• Dimensional viewpoints
• In your explanation, you show a transference of ideas up from the 2D to 3D. How would transferring the 3D to 4D look (or vice versa). Could this line of thinking help break-down the understanding of the 3D co-ordinates.
• GPS already solves something similar to this?
• GPS appears to be a problem with similar facets, using four co-ordinates within a 3D system to locate a fifth position. It would seem that approach uses a semi-octahedral solution. Perhaps look at that solution for inspiration.

In thinking about this I mostly realised that naming conventions tend to be relative. I may be walking forward on the planet, but what direction is that relative to a moving position in the Andromeda galaxy. I like the idea of what you are doing, hopefully this will inspire something in your chain of thoughts.