You are correct. Using the engine offloads or minimizes harmful cold battery draw-down for driving and heat.
My experience suggests "both of the above". One of the factors is the amount of cabin heat requested by the driver. If the ICE starts immediately in cold weather, I can trigger a transition back into EV mode by turning off the heat (after the ICE has run long enough to satisfy its need to come to temperature). I suspect that the battery temperature also plays a part in the decision, as the car's controller is sensitive to the battery efficiency and how much effect the resistance heater will have on how much power it can supply. On the other hand, it's all interrelated. If it wasn't cold, I wouldn't be requesting power for heating the cabin, and the battery wouldn't be cold enough to affect its efficiency. The car is just too smart.