What's Up With Saving?
Horror games try to bring you into the experience as seamlessly as possible. In House of Ashes, that means there's basically no indication when you move between the game's scenes or when the game is saving. This is great while you're playing because you totally forget it's a video game, but not so helpful when you're trying to quit and you don't know what progress has been saved.
Generally, it's safest to quit when you're switching between characters. It denotes that you're switching perspectives when their name and the depth appear in the bottom left of the screen. You can assume that all your decisions with the previous character were saved at this point. If you really need to leave in the middle of a character's session, you'll still probably be fine. At most, you may need to repeat the last decision or two.