Can one just keep a picture/scan of the Hong Kong landing slip, or does one have to keep the paper version of it?
Hong Kong landing slip:
You don't need to keep the landing slip at all, it's for your own reference only. If you use the automated gates (mirror) at the airport, you'll never even talk to a human when leaving, much less need to show it to anyone.
The one catch you already pointed out in your other question: you may need to show the slip if applying for mainland China visas in Hong Kong. (And no, I don't know if PRC authorities would accept a photo.)