My son's Cobalt SS won't start: battery?

Hi!

I just registered because my son's 2006 Cobalt SS won't start. He bought it before winter, used it a little (like a month), then parked it in the backyard and didn't start it since a while.

Of course the battery is now dead. I will replace it but I'm not sure this is the only problem: when I jump start the Cobalt with my 2020 Hyundai Elantra, the car starts quickly and everything is ok.

BUT if I unplug the Elantra battery and plug it into the Cobalt, then I can't even turn on the car (so of course I can't even 'try' to start). Looks like there's a short somewhere (maybe not, but it looks like there is).

My question for you guys who know the Cobalt very well, is: is it because of the type of battery the Cobalt SS uses? That would explain why it won't even turn on the car with my Elantra's battery?

Or is there something else I should look at before I replace the battery? I'm quite confused right now...

Thanks for your help!

(I'm attaching a piucture of my Elantra's battery, in case it helps...)