iPod Touch 2nd Generation (Used) boots into Apple Logo when dead, then shuts off. If charged however, everything is normal. What is going on?
"You do realize this device is almost 10 years old, and that the battery is probably showing its age? I suspect that any amount of calibration you attempt isn't going to make much of a difference and will continue to present you with perplexing results." - IconDaemon

"The firmware in the 2nd generation might not be as advanced as what's found in the iPhone 5. In addition, the 2nd Gen Touch can only take iOS 4.2.1. Perhaps a combo of these two things, in conjunction with an old/failing battery, may be causing this anomaly. I don't think you'll get a satisfactory conclusion to your quest -- it may simply be an unusual set of circumstances that is leading to what you are seeing. Lastly - it doesn't seem as if the 2nd Gen is malfunctioning in any other way. In the immortal words of Lennon/McCartney - Let it Be." - IconDaemon

The battery meter will display unpredictable fluctuations in the percentage due to the battery being very old, causing the device firmware to be unable to accurately measure the percentage remaining.