On the show I was a bit suprised to find out people think "shut the light" is an unusual expression. I've used it all my life and no one has ever commented on it. I notice I use it a bit differently than "turn off the light". I would say "shut the light" when the result would be darkness, not meaning to turn off a desk lamp, for example. This is analogous to shut (off) the TV or the car, or the fan or the burner or the faucet. Maybe these are also unusual to people?
https://waywordradio.org/discussion/topics/shut-the-lights-a-regionalism-or-what/ is another thread on this topic.
I use "turn off" most. Now that's an anachronistic verb phrase! I would use "shut off" or "kill" (in some contexts) without thinking twice. I have heard "shut the light." I might have used it rarely, in a distracted moment.
For me, there are slightly different contexts for each of the phrases, while all are possible in any context. I would more likely use "turn off" when speaking of a single fixture or switch. "Turn off the light on the table." For multiple lights, I would more likely use "shut off." Shut off the living-room lights. To emphasize complete darkness only might I use "kill." "Kill the lights in the basement."
This got me thinking about the use of "turn" to describe controlling a light. Especially since the mechanical action is better described as "flip" or "toggle." And these days, with the new touch-sensitive controllers, perhaps "touch" or "tap" is a better description. And no ... I don't want to include "clap."
Ngrams shows this interesting result for the phrase "turn off the light."
Makes me wonder if the use of "turn" dates to the time when kerosene and gas lamps started replacing candles for nighttime illumination. With those devices, one would literally "turn" a control to retract the wick or shut off the flow of gas. Of course, you couldn't "turn" the light on without the assistance of a flame.
Also, just to stay on topic, I never use the phrase "shut the light." Don't think I've ever even heard it before. Grew up in the Midwest, now living in Arizona.