I think that flu thing is an old wives tale. You usually get flu because you breathed it in. The association with cold is because during cold weather people spend more time in poorly ventilated areas.
My experience with exerting in the winter is I start feeling hot around the neck and upper torso first. So if I'm wearing a thermal jacket below a windbreaker, I'll start with leaving the thermal zipper down a bit but have the windbreaker's up all the way.
Problem is, some take way longer to heat up from exercise.
I will usually not reach a comfortable warm until 1-2 hours in doing regular walking (and no, I will not do a 250m sprint to finally feel warm).
For everything else: Agreed. For cycling in the cold I recommend biking gloves. Game changer!