There is a known issue with battery indicator on 602 that should be fixed with one of future firmware upgrades.
First of all it doesn't show green light after 10-12 hours of charge. Second, when you turn the device on the battery indicator appear empty, but when you start using the reader (opening books, turning pages, etc) the indicator adds one battery charge square after another until it shows the full charge.
The problem is with firmware, not the device or charger.
Mine 602 behaves exactly the same way and I am using it for a second week already (didn't need to recharge it second time yet).
In addition I would recommend to install the latest version of firmware 2.0.2 (if you haven't done it yet) from
here.