How long did you wait for the charger to charge the battery?
Looking at CoconutBattery 1st snap it does appear the battery IS charging! You do need to give it a good 5~8 hrs to fully charge as the battery is so low.
The battery charge is gated by the thermal limit of the battery so the charging won’t be at the full rate throughout the charge. So far the battery is not that warm so it doesn’t appear to be charging a long time. In addition, it does take CoconutBattery a bit of time to be updated by the OS and the OS updated by SMC so don’t be to quick jumping you have a problem.
So I would shut the system down and let the system sit a good five hours under charge at which time the MagSafe cord LED should change from Amber to Green. Then take a fresh snapshot of CoconutBattery and post it here so we can see where you are.