Knowing that you have a 3.0i (aka M54B30) engine helps. The engines have different physical layouts, obviously, but more importantly, we all have different experiences with our E53's and where I may be able to help with your M54, I wouldn't be much help with an N62...

You did the best thing shutting her off when you saw the temp climb. These M54's HATE high temps... quickest way to a blown head gasket, or worse...

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2003 E53 3.0i
DOB 10/15/2002
Sienna red, Savannah Beige Dakota leather
Type FA53, Engine M54B30

2018 G02 30i
DOB 4/21/18
Black Sapphire Metallic / Canberra Beige Vermasca leather
Engine B46O