I was assuming that Ledger would display my BTC balance on the screen.
It does. Not on the Ledger LCD display (hardware device), but on the wallet desktop software that controls it (Ledger Live).
So how to make sure my money in on cold wallet device? From the moment I see money on the Ledger Live, does it mean it is already on the device?
Just check your balance with the Ledger Live. Or search your address in a blockchain explorer. If the coins are there, they will show up in the balance.