Tire and car manufacturers are aware that tires will heat up during driving. The "spec" tire pressure (door frame, or owner's manual) is what you should use, year-round. It's based on the tire being at outside-air temperature, not on tire temperature after driving. This means that you will have to add some air as winter sets in, and bleed a bit out in spring (if you're obsessive about having the right tire pressure). If you're checking your cold tire pressure when it's ridiculously cold out, you might well get a slightly low reading that you can ignore. But if the outside air temperature is normal winter temperature and you get a low reading, add air.
There's not enough difference in temperature to expansion to be a problem. If you get a low reading during cold temperatures you should add air. BTW the proper tire pressure is listed either on the B pillar inside the front door opening, on a sticker in your trunk or in your owner's manual. The proper pressure is based on your car, not the tire.
The tire pressure doesn't change that much depending on a temperature. If the manual says check the pressure when the tire is cold, that's the right way to do it. If you measure the pressure after few hours of driving it won't be too different.
Cold means cold, as in not recently driven on. Coldest time of day (morning) gives you the best baseline. Tire load carrying capacity diminishes as air pressure is reduced. You want to set them so that the recommended pressure is achieved at the coldest temperatures the tire is likely to encounter.