Vtyjkyuk

What do they usually write on the sign in public places to inform the visitor that the service is temporarily paused for the lunchtime? "Lunch", "Break", "Out to lunch"?

(Can you, please, specify which English-speaking country you mean?)

In informal conversation, it's common to say "I'll be out for lunch from 12 to 1" or "We break for lunch at 11".

A sign might say "out to lunch", but I think this is rare today. It's considered too informal. Usually a sign will give the hours, like "Open 9:00am-1:00pm, 2:00pm-6:00pm". The idea that the time from 1:00 to 2:00 (in this example) is a lunch break is usually not stated but simply implied by the gap.

A sign wouldn't normally just say "lunch", as that would not be clear whether it meant that the place is closed for lunch, or that they are serving lunch, or just what they are doing about lunch.

There isn't any official sign that I'm aware of in the US. But we wouldn't simply say Lunch or Break. We Americans are a bit fussy about putting things clearly in any sort of official communication, often to the point that those communications are so wordy that they are hard to make sense of.

We might say Out to lunch or On lunch break, but that doesn't cover other reasons why we might be closed, and nobody really wants an explanation anyway. They just want to know when you will be back. So more typical is a sign that simply says when you are going to return.

There is a popular sign says WILL RETURN and then has a clock underneath with movable hands. If you'd like to see one, you can have a look here. I see this sign often.

"Closed for Lunch" is what I would expect to see. Google image search for the term backs this up.