Car not starting until battery is reconnected

First time posting here. I just bought a 2013 Prius C with 68k miles last week, and drove it for the fourth time since purchase today. I really like the car and there have been no issues with it until this afternoon. I ran out to do some errands and the drive was uneventful, but when I went to turn the car on and head home all of the warning lights on the dash came on and the motor/engine wouldn't start.

I did some looking and found a lot of threads saying it's likely a battery issue, so my father drove over with a multimeter and jumper cables. We checked the battery and it was reading 12.5v, tried starting it a few more times and still nothing, then tried jumping it to be sure and again no change. We then decided to see if we could get the battery tested and, if need be, replaced at a nearby auto shop so we disconnected it, and just out of curiosity we reconnected it to see if anything would reset or change, and sure enough the car started the second time we tried the ignition after reconnecting it. It looked like someone had been in the battery compartment before as there were some tabs missing from the plastic below the bacl seat, and there was a little bit of electrical tape around the wire to the negative terminal which my father took off. On the way home it ran just fine, but it did give a short beep about five minutes after it started while I was waiting at a red light.

I am at a loss as to why this happened, so any opinions would be greatly appreciated. I have very little knowledge of car electrics, and my father is similarly confused by the issue. I'll likely end up taking the car to a dealership and have them look over it if there's nothing I can clearly tell is wrong, but anything to point us in the right direction would help immensely.

 

Thanks for the reply, sorry I couldn't get back to you sooner. There are no lights on the dash now and the car appears to be running normally. Exact mileage is 68,403 and I have only put around 50 miles on the car since purchase. I took a look at the coolant reservoir like you suggested and I could not see any noticeable movement, and the lines to and from the reservoir were not warm to the touch after running the engine for about half an hour, but I could feel a bit of vibration so they may have had some flow.