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A lot of the plot points you described sounds like "Hide and Seek", but that was just a man and his daughter. There are a couple scenes with an overflowing bath and writing on the wall, the daughter has an "imaginary friend" who seems a lot like a demon and who stays invisible for most of the movie, and the daughter does a lot of creepy things including drawing weird pictures. There are also scenes in the woods near their house, and also in a cave. But again, there is no son, just a man and his daughter. First scary movie I ever watched, and it freaked me out badly! Haha!
Edited by gaimanite.pkat

A while back I saw bits and pieces from this horror movie that I cannot for the life of me find. It was about this family who I believe lived in a cabin (or at least a woodsy area). There are only three things I distinctly remember: the boy has an "imaginary friend" (who is undoubtedly a demon), this demon overflows the bath and starts writing in black water on the walls (and no, the film I'm thinking of is not Dark Water), and the a friend of the father's asks his little girl what she's drawing, and she says it's a picture of him, but he's dead. It probably came out in the mid 2000s (2003-2007).