In effect you have discovered that you are driving an e46!
All of them operate the way you described.
The OBC temp reading is accurate, but the temp gauge is not actually a temperature gauge, but an idiot gauge. Might as well have an idiot light instead.
The gauge is not analog, but in effect operates in damped steps. That's why it's not a great thing to use to monitor engine overheating when you blow your ET or whatever else. It lags while you relax and let your head overheat and warp.
Good going BMW.
I believe the sensor in the hose is the temp sensor for the electric cooling fan. So I'm been told.