Which Power Adapter is the correct one for Powerbook G4 15inch FW800 - 45watts or 65watts
In practice, I've never seen a properly working 45 watt adapter not be able to boot a properly working PowerBook when you remove the battery.

The 45 watt adapter was intended for smaller devices and would often not be enough to both charge and run a 15 inch PowerBook in all situations even when everything was working properly.

At this point, with ten year old equipment, it's possible you will need a 65 watt adapter or have some power management issues with the adapter or the Mac.

That Mac should have a 65 watt adapter for best results, power on and charging since the G4 and battery do need that much power to work as designed.