As a native speaker of American English, I agree with the original poster's interpretations. "Turn into" usually means "transform into", as in the original poster's two examples. "Water could turn to ice" sounds good to me; this happens to be a very poetic example. "The magician turned the cat to a rabbit" does not sound as good to me as "The magician turned the cat into a rabbit", because it is possible that the magician might be turning the cat to face a rabbit.
"Turn to" can be a poetic phrase that means "turn into". It is commonly used to mean "orient toward" or "rotate toward", as in "The cheerleaders turned to the right." It is sometimes used to mean "choose a different option", as in "Frederick the Great turned to England for help."
"Turn in to" usually means "rotate toward", as when a driver "turns in to the driveway".
"Turn in" can mean "submit <something>", as in "The students turned in their homework." "Turn in" can also mean "go to bed", as in "I think I will turn in for the night."