Someone in another thread talked about not charging before you leave in the morning, rather, charging when you get home in really cold weather.
Yes, I've heard that suggestion, and it's got some merit but not for the reason you suggest:
This was to charge the battery when it is warmest to prolong its life.
It's not to prolong its life.
This assumes the car doesn't warm the battery when it is plugged in before charging.
...which is false. That's why this isn't about prolonging its life. The battery management system absolutely track carefully when it can charge and how fast based on how cold the internal battery temperature is, and it
will warm it up to make sure it only charges at a level that is safe for its temperature.
The real purpose for charging right when you get home is because of that factor. The battery is already warm when you get home. You can have it charge while it is already warm, and the car won't need to run any battery heating. Otherwise, if you get home, and leave the car to sit for the next 5+ hours and get cold before its charging cycle, then it has to waste energy to run battery heating to get it up to a good level to do your charging. It's simply about not wasting energy if you can just get it charged while its already hot from driving instead of letting it get cold and have to re-heat it.
I know the charging rate must be reduced when the battery is cold. I don't know what the car does if well below acceptable temperatures for charging. Does it warm the battery first? Does it limit the charge rate to a trickle?
For those last two questions, Yes. & Yes.
There is a level of coldness where it absolutely won't charge. It would be severely damaging, so the car just prevents it when it is that cold--period. So, it must run battery heating to get the battery pack warm enough to safely start any charging at all. But that is not an instant crossover point from nothing to fast charging. It is a gradual scaling up with temperature. So once it gets warm enough to start, it will start with a low safe level of current while continuing to run the battery heating, using that incoming power from the charge cable. So between the battery heater and the act of charging itself creating a little heat in the cells, it continues to get warmer and keep increasing the charging current.
One thing to be aware of, if you are on a trip, the charge rate can be severely reduced if the battery is cold. They say it will heat up if the car knows it is headed for a charger. But the only way to tell it that seems to be to set the charger as your destination. I've entered destinations where I would only have 5% remaining and it didn't even suggest that I stop and charge. In that case it would not warm the battery on route. Teslas are strange and very unpredictable.
I've gotten awful recommendations from the car Nav like that if I let it choose Superchargers for me, so I just don't. It seems to have a parameter set to prefer the least amount of stops, and it will do ridiculous things like extremely long charge times and tiny arrival % to try to skip over Superchargers to meet that criteria. It sucks. So I just pick the next Supercharger a reasonable distance along the route, that can have a normal arrival % like 15 or 20, and that works pretty well.