Vista internet connection issue, can't reset adapter

I have this Vista computer of a relative. Here's the issue, I take the Vista desktop home with me, connect it up to my Linksys router connected through AT&T DSL and opened up Internet Explorer & Firefox and EVERYTHING is PERFECT. I can go to any site at all.

So when connecting to their DSL modem direct, and accessing the settings through modem, I click on "diagnose connection" and EVERYTHING passes, the PPPoE, ADSL, etc. passes. ALL GREEN. And then when going to "Login", it clearly states, "You are now connected to the Internet."
But yet, IE & Firefox CAN NOT access Internet, I checked firewall and even disabled it all completely, still can not access it. I am able to ping the local loopback and also ping yahoo.com and google.com, & other sites, but still doesn't not work. So taking it home, I know it can not be the firewall, or TCP/IP stack, or anything wrong with the computer period. It connects fine at my place.

Going to the Network screen and seeing the lines from the Network to the PC and to the Internet, there's a RED X on line from the NETWORK to the Internet saying CLICK HERE to diagnose, so I do that and it is unable to find the problem, but states there is a problem and then it says to try to reset the adapter, but yet that is unable to solve problem too the Vista OS states.

So I am believing it's either on their end, ATT or with the wires in the home. I tried another brand new ethernet cable, but same issue. When I click on the right bottom in the taskbar icon, the internet icon states it has "LOCAL ONLY" connection, sometimes it says Internet and it goes away every now and then, but even when it says Internet it still can not access it. So can someone please tell me what else would be the issue. I appreciate it ALOT. I also checked the Network location, I put it both on Public & Private, same issue. same problem. I thank you everyone for those who help.