Does anyone say “press the light on”or “press the light off” to mean “press the button to turn on/off the light”?

I’ve never heard this in any dialect. “Press the x on” sounds strange for any noun unless you are physically pushing an object onto another object to make it stick or something like that. In the USA: Turn on the lights (by far most common) Hit the slight switch Flick the light switch Hit the light(s) (turn on) Kill the light(s) (turn off)

Not that I have ever heard. “Turn the light on/off” or “switch the light on/off” or “flip the light on/off”

I could maybe see this being used if it was one of those lights where you actually push the light itself to toggle it on off (e.g. https://byjasco.com/media/catalog/product/5/4/54807.ep.00.jpg) but it really doesn't sound natural to me. "Turn the light on/off", or "switch the light on/off" are by far the most likely wordings I would expect - or slight variants on those like "Turn on/off the light".

press the X on/off is not a valid phrase in English as far as I am aware

In English English it would nearly always be "turn the light off" or "switch the light off".

No. A lot of learners ask this, because it's the way they say it in their language, and it's logical, but it's still no. No English dialect says press the light on. "Turn the light on" is the most standard way. It's also the kind of phrase that's a bit confusing because it does work as a sentence, even though it doesn't convey the meaning you intend. It could be misunderstood as a short way of saying "press the light that says on." Or even, if you're working in a hotel and talking about small lamps, it could sound like you mean "touch the light to turn it on." Some lamps are touch-controlled, after all. So it's not just incorrect, it's a mistake that you should avoid making.

No, if I heard someone say that I would instantly know they were not a native speaker because it it sounds very strange.

Even in my grandmother's house back in the day, where there were actual on and off buttons on the wall for the lights, we never used "push" or "buttons" to talk about turning lights on or off.

Most people don’t have buttons for their lights, they have switches  Like this. No button to press.

Switch, turn, or flip. Also hit, but only for turning the lights off.

No.

No, it would sound bizarre because using the word “press” emphasises the physical motion, when what you’re trying to express is that you made like light go from “off” to “on”. Additionally, it’s not the light that you press, it’s the button, so there’s an omission like “press [the button so that] the light [turns] on”. We generally don’t use resultative constructions like that in English. Like if you won a prize for shooting, you can’t say “I shot the prize for myself”.

As others have answered. But in the UK, we literally only say; Turn the light/s on/off That’s all. We don’t say flick, hit or kill, just turn it on, turn it off. That’s all.

No.

The only “press the ___on” that sounds correct is “press the button” ;)

I guess it would make sense if the button itself is the thing that's lit up. Otherwise, no, I wouldn't phrase it like this.