I had those same problems a few months back. It was the EGR cooler. I installed an EGR delete and a new oil cooler.
My oil cooler was so clogged water from the garden hose barely passed through it.
Luckily I did the oil cooler even though I didnt think anything was wrong with it. I dont know if the oil cooler failing could cause your coolant to purge from the degas bottle. I think it would not though.
What happens is the oil cooler starts to get clogged on the water side. The coolant is restricted when this happens. The coolant that exits the oil cooler then gets directed into the EGR cooler. With this coolant being restricted and flowing much less than normal it allows the EGR cooler itself to get overheated. When this happens the insides of the EGR cooler tend to crack and allow the coolant to dump into the exhaust side of the EGR cooler which feeds into the INTAKE and into the combustion chamber.
When the EGR cooler first starts to go it is just a seeping of a small amount of coolant. Over the course of say overnight, there may be a shot glass or 2 worth of coolant that has seeped into the exhaust side of the cooler. When you now start the engine this coolant is burned off through the combustion chamber which may take a few minutes. That is why you have white exhaust coming out til it warms up. Warming up actually has nothing to do with it, it is just a coincidence bc thats how long it takes to get burned off.
If you have an Edge Insight CTS you can monitor the EOT and the ECT. These temperatures should be close. Over a 14 degree or something like that difference it would tell you the oil cooler is clogged.
White smoke could be fuel burning off as well, just depends on what the smoke smells like.
I hope that helps even though I did not directly answer your question.