Coolant burning has a very distinctive odor...somewhat sweet. If you can't tell, any trained mechanic should know what burning coolant smells like. And there's NO way any part of the cooling system external to the engine (like the prone-to-failure overflow container), can cause white smoke in the exhaust. Assuming it IS coolant, there are a few spots where this can happen, for example a blown head gasket. (Recent oil change guys put in too much oil???)

N62's are also known for making oil smoke in conditions like you describe; often quite white in color, (somewhat blueish??). Often the CCV is gummed up, sometimes valve guide seals ($$$).

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2014 BMW 328i Xdrive
2011 BMW 335i M-Sport
2008 BMW X5 4.8i Sport
2000 BMW 528i 5sp