"Connected" definitely refers to the Bluetooth connection to the car. If you turn off Bluetooth on your phone, it will no longer show connected. My phone (Samsung S9) can connect from 100+ feet away, but that alone is not sufficient for the car to be unlocked. It requires me to be closer (less than 20', but I haven't tried to narrow it down further).
no, it's the phone key. if you open the app, the only spots that say anything about "connected' or "disconnected' is the phone key. also, my phone right now says "connected" in the notification and the app, but the actual bluetooth connection to the car (that would be used for phone calls and audio and such) is not connected. they're two different things.
at the same time, of course bluetooth has to be turned on - the phone key uses BLE. if your phone's bluetooth is turned off, the phone can't connect to the bluetooth receivers to use the phone key. you can be standing right next to the car with the phone key saying "disconnected" and it won't let you unlock, regardless of how close you are. ask me how i know.
It seems having your phone connected may hamper the car's ability to sleep. When I first got the car (FW 28.3), it seemed the car was losing several miles at night even though it was plugged in. The phone was in the bedroom right above the garage. I started turning off Bluetooth at night, and the drain went down to 1-2 miles per night. Might have been a FW issue. Now on 34.1, but haven't tested it.
this may be true about the bluetooth connection itself, but is not true of the phone key. my phone key stays connected all the time in my house, yet the car is sound asleep in the garage at the moment.
one strange thing i've noticed is that if you turn off bluetooth and back on after you've exited the car, it will reconnect to the car and stay connected (actual bluetooth connection, not phone key). i have no idea why it does this, but that might be what you're seeing.