Actually use the epoxy. I use it for floors, you can mix the colors and on a door like that its a lifetime repair. I have had crack repairs in floors at 8 year repaired and counting(in a foyer walked on constantly), still looking perfect. Just squeeze it together, It's a two part. But make sure yours are fresh as if they are too old they are more difficult and not as smooth to work with. This is NOT a floor filler for tons of gaps between boards though, it's for a repair here and there, as your door pics show.
I use these every day. I have a couple of the full kits always on hand and mix them with other brands on the market with perfect results. When I make a repair and mix the colors for a color match it's very difficult for me to find where I made the repair, sometimes I can find the repair at all. This stuff is durable, hard as a rock, can be left as is or worked and coated, its up to you and the situation. Sometimes I mix it with saw dust, most times I dont.
Caulk is crap for what you are doping, this is the ticket. Now if I was trimming out a house with poplar, yeah I would caulk the raw wood before priming and painting, but NO Caulk on a kitchen door miter, that's a mistake. Use this stuff or make your own putty with epoxy, saw dust and tint if needed.http://www.mohawk-finishing.com/catalog_browse.asp?ictNbr=113