How do I repair this interior door?
I would try applying some wood glue  into the split with a small brush. Sandwich the door between two 2x4's with some clamps. It may help to put some wax paper on the 2x4 to keep any glue from adhering the board to the door. Tighten the clamps until the glue starts to squeeze from the crack. Let it sit 24 hours, remove the clamps and scrape any excess dried glue. Repaint to hide the repair.