Hey akobak
This is a pretty interesting issue. I'm really sorry you haven't been called yet, but hopefully we can figure it out before you do get a call and beat advanced support to the punch.
So you can get to the System Control Center, which is good. How do you access it - with the IP address of 192.168.0.1 or
http://systemcontrolcenter.com? Do both of those options work to get to the control panel? Our engineers actually made changes to your access gateway (where your 'net comes from) so I wonder if it could just be some DNS mismatch going on. Upon inspecting the LAN connectivity of your modem, it seems that information does move back and forth so I don't think there is an issue with the LAN port or the cable.
What websites are you trying to access when you test out connectivity? I know you've probably done it, but I assume you have powercycled both your modem and your Mac? Any other devices in the house you can test?
Looking forward to resolving this with you
Amanda