When I boot an Android device into download/ODIN mode, on the top-left it says "CUSTOM BINARY DOWNLOAD (
n counts)" where
n may possible mean the amount of times the device has been flashed. Is this correct?
Yes, you guessed it right. It increases everytime you flash a ROM / kernel. It's also called binary counter. It's one way of checking if your device has stock ROM or not
The binary counter in a Samsung built device is a flip counter which turns up or increments a number after a non-official ROM is Installed.