How can I fix a hairline crack in a thin wood panel of an exterior door?
Lightweight spackling compound is not suitable for exterior use.  Even if painted, water will effect it greatly.  In addition, it is prone to cracking as it is brittle and probably will not withstand opening and closing of the door (vibrations).

Wood filler or wood glue may work to fill the crack and would be better for exterior use.  The glue is probably more flexible than the filler and may be better for filling the crack.  I've used the wood filler before, but usually to fill nail holes or fill in small cracks.  For a few bucks and some sanding, it may be worth a go round.

Another option would be to caulk the crack.  Since caulk is flexible, it will resist cracking better than the wood filler.  Both caulk and wood filler would be paintable.  I don't think you can stain caulk, but I believe the door is painted on both sides.   

The other option I mentioned in the comments is to replace the panel/door.  The panel is most likely held in by surrounding trim, so if that can be removed, just the panel could be replaced.

Another option would be to get a very thin piece of veneer wood, cut it to the dimensions of the panel, and then glue it on top of the existing panel.  Since you cannot see the crack from the outside, this could be done on the inside and then painted to match.