Yes, the phone has to be connected to the internet for it to work. It says so in the documentation, and on the very first page "To use the Signal desktop app, Signal must first be installed on your phone."
You have confused me. The statement you quote seems to say that Signal must be install on the phone in order to get Signal working on the desktop. That does not, to me, imply that Signal must be up and running on your phone while you use Signal on the desktop.
Suppose I install and configure Signal/Phone, then install and configure Signal/Desktop, then turn off the phone.
Will Signal/Desktop still work while the phone is turned off?
* If yes, then what you say appears to be wrong.
* If no, then that's genuinely astonishing to me, and I've learning something.
The previous thread is incorrect. Signal desktop can work despite having the mobile phone active. That's the key advantage over Telegram and WhatsApp. Telegram's secret chats are not synchronized. Neither is WhatsApp's, and even worse, WhatsApp's desktop-based chat is just a projected copy of mobile. In other words, WhatsApp's desktop version works as long as mobile is active/is connected to the internet.
This is incorrect. Signal desktop can continue to work without having Signal mobile active. My mobile phone is on airplane mode and is being charged at the moment, but I am on a call with my colleague using Signal desktop. They are on Signal mobile.