After seeing the latest posts here, I did a couple of tests and found that if I don't use the APU, or EP (External Power), this still happens (same as my original post)! To be honest, without using either of them, the battery should drain, especially if you turn everything (Window Heat, Galley, Nav Lights, etc.) ON like I did.

When I use the APU, or EP, it no longer happens. I left the 737-200 sitting there for more than one hour using the APU, or EP, and then I started the engines, and did a a short flight. The 737 performed fine. So in my opinion, this problem is fixed. ;)