For vintage pre-Intel based Apple Portable computer power adapter identification see:
•PowerPC-Based Mac notebooks: Identifying the right power adapter and power cord--US
The battery, power adapter, and DC-in board all work together to alternately power/charge
the computer, and charge the battery, also to let the computer run at correct voltage when
the battery is fully charged. Or to run the computer at reduced CPU without battery installed.
These three component products need to be correctly functioning for this to happen. Also
there is a fourth possibility outside of these three, in this line of thinking, that may contribute
to a failure of the system to function as expected. The logic board can sustain some kind of
damage that may cause the charging/power systems to fail, outside of the above component
parts that may test OK otherwise.
The DC-in Board issue can be a part of a greater level of troubleshooting by replacement
of parts, however difficult opening the iBook computers can be, to be sure the logic board
has no defect which may have occurred due to rough handling.
Then, the other issue with the logic board and graphic chip, involving Shims to jamb the
failed solder joints together, could affect some models among the iBook G3 & iBook G4.
There had been a few articles online by a guy who'd figured out how to shim these years
ago, and I am not sure of his name (Cory? something) but a general search such as this
https://www.google.com/?gws_rd=ssl#q=ibook+solder+defect+repair+shims does bring
up V-chip repair by shims, and may touch on the re-flow or re-bead of cracked solders.
Hopefully you can make your old iBook G3 work OK; and that the logic board or graphic
chip, etc are also not part of the issue.
Good luck & happy computing! 🙂