How to go about fixing this window that won’t pull shut, either temporarily or permanently? So much cold air is leaking in

The pin you see is the part that secures the handle to a shaft that sticks out of the window latch assembly.

To get the handle working you are going to have to carefully observe how the moving parts of the good window work and then compare that to what you see going on with the bad window. This may include activating the handle back and forth while the window is open. Look from below, outside and the edge end of the window. Doing this you may notice that something has blocked the mechanism and is keeping the latch from working.

It is possible that the latch mechanism is just flat broken in which case either it or the window would need replacement. Many times these latch mechanisms have nylon or pot-metal parts inside which break if forced too hard.

For a temporary fix till you can get the actual window and/or the latch replaced there are number of things you can do cut down on the cold draft through the crack. The simplest may be to simply apply some blue painters tape over the cracks to cut off the air flow. Duct tape of "gorilla" tape are other possibilities too but will take a lot more work to clean up after the tape is no longer needed. There is also a stretchable clear plastic product available that you can stretch over the entire window opening that acts as an inside window seal that can be used in the cold season. This method has been recently discussed here on DIY.

This is the modern equivalent of a sash weight.

Yours is either a (window) spring balance or a spiral balance.

But you seem familiar with the terminology, so the real question is: what's that little plastic part called?

It's either a Nylon Sash Cam or a Shoe Set I think.

It might just be easier (and better) to purchase a whole Window Channel Balance Repair Kit and replace all the plastic parts. They're like $5. Then again, an entire balance costs like $10.