You left out
Here is my take on their distinctions:
shake - First, its a verb. I can be doen to someone or done to oneself. It generally is a willed act, even if it's not a welcome one (meaning some agent is involved).
tremor - A noun, usually describing a subtle but steady "shaking", usually involuntary, often pathological (meaning a symptom of something).
shiver - Very similar to tremor but less acute (usually the whole body or at least a good half shivers). Still involuntary, though there can be an identifiable cause (cold weather, fear, bad thoughts). Usually goes away either as fast as it came on, or if due to strong source, when that source is gone.
tremble - Could be the same as a shiver or a tremor, but if it's distinct, usually describes a shaking or unsteadiness caused by lack of power. You could be trembling when you stand up due to muscle weakness, or you could be trembling at the feet of a king because you are weak with humility, etc.
Overall, the finer points of the words are going to be cause, locality (part of the body) voluntariness, duration, and severity.