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.
Though the crack is diagonal it looks relatively straight. In this case I'd take a piece of wood the approximate thickness of the crack at it's widest, cut it to length and then taper it so that it fits quite snugly - you should have to use a mallet to tap it home. Don't worry about the thickness too much - but obviously it should be fairly close.
Once this fits use wood glue to bed it in and fill the imperfections, though you might get a better finish using wood filler for this. Basically put more glue in than you need so it gets squeezed into every nook and cranny. Clean it off before it sets, then leave the repair to set.
Once set use a chisel to remove the excess from both sides of the panel and finally sand to make it flush with the panel. Fill any remaining (and hopefully small) holes, prime and paint.
By making the fit tight you minimise the chances for it to move, and therefore crack, further.
Best Answer
Hard to say whether it's a tenon or a few dowels, but assuming the wood is sound, you can pull the rail into place with a clamp (or a strap if you're low on 8' clamps) and put a couple of 8-10" screws through the stile into the rail. (Plenty of skinny long screws these days.) Predrill with a slightly smaller bit. Do a bit of a counterbore and you'll be able to hide the screw heads with filler.
And if you haven't, paint the top/bottom of the door to seal it.