I started another reply, and it had lots of steps to try to repair this situation, but then I rethought.
If this is a fresh install, and you have no data to keep on the Debian system, here is a bulletproof solution: Reinstall. When you reinstall, don't do anything fancy with network. Just let it do the default with DHCP. Plug it into the router LAN jack as you do the install. Once Debian is installed, run a web browser and go to the webpage on your router as documented in the router manual. In the router's web site (something like 192.168.1.1), set up the Internet connection to login via PPPoE. If you have already set up the router from your Mac or Windows system and a web browser, you don't need to do it again from Linux, it will just work. If the Debian system can see the router website, but not the Internet after setting up the router with the ISP info, try one reboot of Debian to give it a chance to load the networking since the router had its configuration done. This is the shortest and simplest path to fixing up the botched Linux networking setup you have. Remember, the idea is to allow the router to be your path to the Internet. It will handle everything, and the Linux system only needs to get on the LAN, behind the firewall on the router. Debian <----> Billion Router <----> ISP Modem <--> Internet
