Ha. Home Depot, right? I'm seconding
@fiehlsport. Home Depot makes a habit of selling "Heavy Duty" extension cords that look quite thick. But on the ones that I've actually seen (and I may be missing something here) there's nary a word about how much current they're rated for.
There's a reason for this: Plastic is cheap. Copper is expensive. Tried using a Home Depot extension cord one year with a 2018 M3 LR RWD when it was cold out and got the same reaction: The car sees the voltage drop because the Wire Is Thin, Small Gauge Copper, Which Has Relatively High Resistance, Which Causes A Voltage Drop. And the extension cord gets warm, too, since that teenier wire is dissipating heat instead of passing the energy onto the load.
Teslas monitor that voltage. The thought on the designers' minds: There's a loose connection somewhere causing that voltage drop and, in order to Save The Day And Prevent House Fires, the current draw is reduced bigly to reduce the heat dissipation in the suspected loose wire circuit. Thing is, the car can't tell the difference between an inadequate extension cord and an outlet about to catch on fire, so, there you are.
I've had much better luck by going to Harbor Freight, which makes a living by selling to contractors who, if they really want to run a high-power drill, they don't want a voltage drop that'll kill the drill, circular saw, or whatever. Harbor Freight's extension cords are rated by
current. So, if one gets a 15A extension cord of, say, 20' length, that costs more than the Home Depot Specials because, well, these use a lot more copper to keep the voltage from drooping. And, if one gets a 100' cord, it costs more than 5X the 20' cord because
they have to use a bigger gauge wire to keep the resistance down on the longer cord.
One of those sits in the under-trunk compartment of each car. We've used them; no reductions in current.
And, yeah, the other posters are right, too: The battery has to be warmed to a reasonable level, and that 12A @ 120VAC = 1440W isn't up to the task in cold weather. A NEMA14-50 socket and 50A circuit, or something else that lets one get up to 20 or 30A will punch the ticket.
Only problem with that.. To stay with code in most places, a garage outlet is usually required to have a GFCI. Those are expensive at 240 VAC since they go into the breaker box, I believe. And a decent, high-power (non-Home Depot) NEMA14-50 also is expensive. The two of those, together, cost about as much as a Tesla Wall Connector which has its own, built-in GFCI. So might as well bite the bullet and go for a Wall Connector, either from Tesla or some other vendor.
FWIW, some states actually provide financial assistance for car-charging hardware; New Jersey is one. A friend just got hers installed yesterday (literally) and is going to get a $250 rebate from the State for the work.