I often spend several hours taking photos in the local dog sled races, and sometimes the temperature is from -10 to -20 degrees F. When holding the camera I place the palm of my hand and fingers around the battery compartment area. I wear thin and soft synthetic gloves with a chemical hand warmer on either the palm of my hand, or the back of my hand. I have never had the need to use the spare battery that I carry in my coat's inner chest pocket. So if the battery is fully charged but the camera does not turn on (energize), remove and reinsert the battery and try again. remember that your camera is much like a laptop or even cellphone. Both can have software and hardware glitches.
There are a few brands of the chemical hand warmers I am referring to. I have been using "Little Hotties" natural, odorless, and air-activated ones. Just remember to follow the user instructions on the label, and don't forget that I am talking about very low temperatures (below zero), not above zero.
In relation to camera lithium batteries, it is best or safer to use the ones designed for your camera by the camera manufacture (Canon, Nikon, Sony, and so on). A poorly constructed Lithium battery can destroy your camera, and even injure you. So you save some money and the battery malfunctions: which manufacturer somewhere overseas is going to honor the battery's warranty and replace your camera?