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@sivikvtec Is the charger connected to the car battery? You have the official MB model, mine is just the CTEK model. Mine came with a pig tail that is attached to the battery terminals and the other end of the pigtail has a "disconnect" plug on it where I can attached the pigtail to the charger or quickly unplug the charger from the pigtail/battery (hope this makes sense). I actually have the charger mounted on the wall next to an outlet in my garage near the trunk of my car, and I run the wire into the trunk, and plug it into the disconnect.
I have a charger like that, and yes the light stays on when it is connected to the battery.
The other lights will be yellow when low, and green when charged.
I disconnect the charger from the battery when away from home so it does not have one more parasitic drain...it has enough already!