No, cell phones do not communicate with cell towers while powered down. This is fairly easy to verify, and it's telling that the screenshots used as proof have deliberately had all attribution removed...
As a side note, confirming no connections with cell towers is not “fairly easy to verify”, it requires a fair bit of semi-specialized gear.
> iPhone Findable After Power Off
Tapping it displays a dialog box with a longer message:
> iPhone Remains Findable After Power Off
> Find my helps you locate this iPhone when it is lost or stolen, even after power off
> The location is visible in Find My on your other devices and to people in Family Sharing you share location with.
> You can temporarily turn off Find My network and it will resume when this iPhone is turned on again.
with buttons "OK" and "Temporarily Turn Off Finding".
“You can even find devices that are offline or powered off.“
There’s some technical details there. The key words are “Find My Network”
It's fairly easy by comparison with more detailed questions about cellular traffic like "is this bit of specific data being sent to a cell tower". It does require some RF test equipment and the know-how to use it, but no specialized knowledge beyond that -- in other words, the sort of thing a company selling a RF-blocking device should have access to.
Back in the times of GSM/2g, and the fable 3310, roaming was EXPENSIVE on Europe.
So, before going to Spain, I would turn my Nokia off.
After returning to my country, and well inside the country away from any Spanish antenna, I would turn on the device.
And every time, I would get a SMS informing me of the roaming costs.