How to insulate your pipes to protect against freezing

Insulate the pipes.

You are likely losing a lot of heat into your basement if they aren't insulated. Unless your basement is finished (and therefore you want the heat there) you want to keep as much heat in those pipes as possible so that it gets into the main part of your house. The heater (and other similar equipment like hot water heater) should heat the room enough to avoid freezing or being a drain on the rest of the house, which is all you really need to worry about.

It can also be safety concern if those pipes are running places where you might inadvertently touch them.

This is the answer to your actual question:

"insulating the coolant pipe inside the actual AC unit itself."

As you correctly observed, if the outside pipe "should" be insulated (it should), then the inside pipe "should" also be insulated for all the same reasons everyone pointed out. However, the key is "should" vs "must." For argument's sake, say the outside pipe is 99 ft long and the inside pipe is 1 ft. Then the 99 ft pipe will absorb 99 times more heat from the outside air. So it's very important to insulate the outside pipe because it will absorb a lot of heat. It's not so important to insulate the inside pipe because it's so short it doesn't absorb much heat. Your AC would be slightly more efficient if you insulated the inside pipe -- but you get 99% of the benefit from insulating the outside, and only 1% of the benefit from insulating the inside pipe. Practically speaking, it's not enough benefit to make a difference.