It is working now.
Initially, thinking that I left out something in the BIOS' settings, I contacted Gigabyte and they sent me an update of the BIOS version from F5 to F9c.
I updated the BIOS version and made changes to the settings as before (since the update returned values to default), and it still didn't work, so it lead me to look elsewhere.
In the end I found out that I used my router's MAC IP instead of my Ethernet's MAC IP when sending the Magic Packets. My computer receives them in either case with the packet sniffer (above) when my computer is on, but I guess it didn't reach my computer when it was powered off/sleeping.