Hi CorBee,
I think the IP address has not been assigned correctly during the booting process. Try this command after the booting process is done in order to know the IP address is being assigned: ifconfig eth0. Try it for both cases: with and without the Ethernet cable connected when you plug the power supply.
Also, use this other command in case you are having the issue, I mean, when you plug the power supply while the Ethernet cable is connected in order to restart the Ethernet connection: /etc/init.d/networking restart and see if the issue is resolved.
Have a nice day!
Regards,
Diego.