For these kinds of synonyms (and also just in general for words with figurative meanings), I feel like its worth looking into their original, “plain” meanings. You can always kind of retrace how the figurative meaning came to be from that.
So I’m just speaking from what I feel is right, but
- 切る has a literal meaning of cutting something, so in the context of “to turn off”, it would come close with “to cut the power”
ex. テレビの電源を切る。
- 消す has a literal meaning of “to extinguish”, and thus has to do with lights etc, when you turn them of, you extinguish them.
ex. 部屋の電気を消す。
- 止める has a literal meaning of “to stop something in place” (in certain contexts; its a little broader I feel like), and so when talking about “to turn off”, you could use 止める to say more litterally that you stop something from progressing further.
ex. エアコンを止める。
Others, please feel free to add to this! 