Is this usage universal?
I've not ever heard or seen it used in the UK. People in the UK would mostly say "he gave notice", "he quit", "he resigned" or "he handed in his resignation".
To "keep something on" and to "leave something on" mean virtually the same thing. Either can refer to an article of clothing that one intends to (or is admonished to) persist in wearing, or to some kind of powered machine (a car, a refrigerator, an electric light, etc.) that one intends to or is requested to permit to remain in the running state.
Both of those can also take "off" instead of "on" as a complement to turn the meaning around into a negative.
Please keep [or leave] the light off. I'm trying to sleep.
To "hang on to" something is a bit different. It can mean one intends to or is requested to not get rid of something.
Hang on to your hopes.
Hang on to that receipt. You'll need that if the IRS audits you.
"Hang on" can also be a request for someone to wait.
Hang on! I'll be there in a minute.
Note that using "off" changes the meaning into something that is not the opposite. It means to literally suspend something or someone from something else.
I wouldn't want to hang off the side of a building until we see world peace.
"Leave off" also can refer to a point where someone stopped doing something:
Where did I leave off in the story I was reading to the kids?
And "keep off" can be a warning against trespassing:
Keep off the grass!
Best Answer
I believe the difference is in the implication.
The way I see it.
Putting in one's papers implies that person is leaving their job but they are going to finish up the last of their duties and get things in order properly before they leave for good.
Putting down one's papers implies that they are leaving the job, possibly disgruntled, and getting out of there. Without bothering with what state they are leaving their workspace or projects in.