I too will have to checkout TAN.
I can say from experience that power toys fancy zones is really cool and I recommend it but it doesn't move the windows back. Some programs also refuse to snap to them (voicemeter) or completly disregards the zones on startup(Spotify).
This is all based on a vague memory of a fever dream from years ago but.....
*In windows you have two options for window placment: keep windows on active screens or don't do anything. *You can move windows around with shift+win+arrow keys. *AutoIT and Auto Hotkey programs can identify individual windows. I would bet that they can also read if a monitor is on or off.
Write a script that check whether a monitor is on or off then move the identified windows. There are tons of tutorials for both AutoIT and Auto Hotkey out there and I'm sure someone has already implemented exactly what you are looking for.
You should not be tipping your landlord.
If they are personaly providing any other services to you, such as maintenance, you need to get those clearly itemized and paid separately from rent.
I feel the same way. In the end, this is the english language where the words are made up and none of the definitions matter.