Hi Dirk,
I had the exact same problem - magsafe always wih a green light, no charging, Coconut told me the battery is working fine, SMC reset without result, MacBook only running with attached magsafe. I found this page when trying to find help, thanks to Google, and found the answers of Dan quiet logic, so I tried to get my hands on a new logic board - which is not easy here in Germany, so I started thinking about getting a new MacBook and dumping the 17".
Meanwhile, only by strange coincidence, I found out that Dans answer is - in my case - wrong. I travelled by train, working with the MacBook, when a stranger with the exact same model sat on the same table. We started talking about my problems and my search for a logic board, when he suggested to put his working battery into my MacBook to make sure it was really no battery problem. What can I say: We changed the battery, instantly the magsafe went orange, the battery-symbol on the desktop stated "Charging", and the MAcBook went on running without magsafe attached.
I got a new battery in the local Apple-Store for far less money than the logic-board would have cost. MacBook is up and running without any problem since then..
One more thing: The dead battery was not orginial Apple, I had bought a cheap battery on ebay when the original part showed the well-known problems like swelling up and not fitting any longer. Maybe the recent problem was caused by this cheap third-party battery?
Anyway, maybe this is of help: Don´t trust coconut, check the battery physically.
Kindly, Andreas