Essentially you’re right.
But hybrids are fine in the cold, despite having batteries, because it’s actually a lot more complicated than that. I’m not going to explain in detail because nobody really cares 
With my electric tractor, which is antique and thus uses over five hundred pounds of lead in its battery pack, the thermal mass of the pack is so high that it can take a week to warm up after the weather changes. Right now it’s got condensation running down it in sheets! But in a Tesla (to provide the opposite end of the spectrum) the batteries are lithium ion double-A cells, very light and fast to warm up. It wouldn’t surprise me at all if just charging them gets them to optimal temperature.
We keep the Nissan Leaf plugged into 240VAC on winter nights, and program it to warm up the car, seats, steering wheel, and battery pack before it gets driven to work in the morning. This works very well, but range is still noticeably decreased in winter due to the inefficiency of electric heating.
