Magsafe Adapter Issue or Battery Issue?

I have a client that has a Black MacBook. I am not sure exactly what model but its definately not a new generation model.

He explains that when he plugs the Magsafe adapter into the MacBook the light stays green and the battery never achieves a decent charge is any charge enough at all to sustain itself disconnected from a power source.

At first thought I assume ok, maybe his battery is done and needs replacement but there is a couple of catches...

He explains that the Magsafe adapter has some physical issues in which I have a picture. My concern is, while the physical wear is unsightly I don't necessarily think that its hurting the Magsafe adapters performance.

Another quick catch is that one of my Apple technicians says that typically the Magsafe adapter is only green when when the battery is fully charged and stays amber color until fully charging the connected battery. Is this true?

Either way, we are unsure wether to change out the battery or the magsafe. ??? I told my client to wait till I had more answers, but also informed him that it wouldn't hurt to replace both as I think his battery is dead and the magsafe, based on the pic he sent me, won't last for long anyways.

Any thoughts?