How do I fix damage to the skin of an embossed hollow-core door?
I would seek to avoid affecting the embossed grain, which will only make the appearance worse. Gently press the upper side of the split to see if you can get it to level out. If necessary, use a razor knife to trim away the interfering fiber material, allowing the surface to come back into alignment. 

Then I'd rub the crack with wood glue to bond it together. Let that dry, then apply a wood filler in the remaining void. Before the filler dries, rub it out with a cloth so that the only remaining material is in the crack itself. Remove all material from the painted surface. 

Finally, touch up the paint as minimally as possible. Use a small art brush and only paint the damaged line. Wipe any paint off the undamaged surface. 

As a rental property owner I'd be satisfied with such a repair done well. It's expected that minor damage will occur in a residence, and this is about all that can be done short of replacing (or re-skinning) the door.