“Turn out” sounds very archaic to me. Still US English, but archaic US English. “Turn off” is current (heh! ) native standard US English. Any other usage is ESL or regional idiom or …
To me at least.
This is a point that has always annoyed / confused me even as a kid.
To “open” a switch is to set it to “off” and block the flow of power. To “close” a switch is to set it to “on” and enable the flow of power. Which of course is exactly backwards from how we’d describe a fluid valve like a faucet or hose bib. There you “open” the valve to enable to flow of fluid and “close” the valve to block the flow.
I’ve never liked how “open” and “close” are applied to switches, but that is totally the official way the terminology is used. Which is why I totally avoid using that terminology.
If I had to guess what “open the light” or “close the light” meant in another language or in an ESL speaker’s idiom, I’d expect “open the light” to mean turn it on. So describing the electrical switch’s action like a water valve, not like the terms properly used for switches.
For me (northern Ohio, with western Pennsylvania influences), the current state of a properly-functioning light is always “on” or “off”. If a light is “out”, then it’s not functioning properly, and is staying dark even when the switch is flipped (which might be due to the bulb being burned out, or might be due to a power outage, or might be a problem with the wiring in the walls, or whatever). The one exception is the two-word imperative phrase “Lights out!”, which sounds slightly more natural than “Lights off!” (and there’s no parallel command to turn the lights on, because nobody needs to be commanded to do that).