It will not turn on with full battery charged

During charging, RECHARGE indicator turns red, when the indicator is fashing,

that means battery is almost full and turns OFF when fully charged.

There's some Manual sites here...
https://duckduckgo.com/?t=ffab&q=PWMA1050BT+low+battery+blinking+red&atb=v159-1&ia=web

Even a modest vehicle battery cannot lose a full charge in 20 seconds. That rate of discharge would seriously overheat the internal connections of the battery and render it useless as well as causing lots of smoke and heating of whatever was causing the dead short.
Clearly the battery does not have a full charge or if it does there is a fault.

The battery clearly needs charging and testing off the car to determine the state of health. I speculate it has a faulty cell.

does the engine turn over but not start or does it not turn over at all

The laptop is the problem and not the battery, check the battery contacts for corrosion or consult a specialist on how to fix your laptop.

replace the battery with a new one. Many battery types create a 'memory' when used repeatedly without a full charge. stopping the charging process prior to full charge lowers the level the battery 'expects' a full charge to be. Do this often enough and you will create a battery that expects full charge to occur at zero charge.

Sometimes the battery will be good enought to run lights but will not turn car over. I would have battery put on a full charge and see if it holds up. if battery will not hold a full charge it is bad. If your car runs and the battery last a few hours then goes low again, It most likley is the alternator. put battery back on full charge and replace alt, and you are ready to run.

When your lg lsp 350t is fully charged, it stops reading like it is charging. You can also turn off the phone and plug in the charger, once it is full, the battery meter will turn blue or show battery full. \but when the phone is on, the way you will know that it is fully charged is that the battery shows a full bar and does not charge anymore.
Hope this information helped.

your battery is broken you must buy new battery (very good battery is 2200 mAh because battery install in your iriver is 1350 mAh)

you simply fried your battery by overcharging it. Just get a new one, let it fully charge and fully die twice before you start normal charging. Then you will want to only charge it for about 2 to 3 hours or find a charger that has a smart chip that tells it when to stop charging once the battery is full.