Comparable description of the issue
The solution I got was too simple not to post.
With battery the macbook (A1342) didn't boot and magsafe connector showed a very faint green led.
After disconnecting the battery, I could get the Macbook to boot. No magsafe led visible.
I guessed the battery (status: service required) had finally shorted internally causing the dim led. I ordered a new battery: Nope. Same problem.
The issue finally proved to be the Magsafe port within the Macbook. It had shifted a tiny bit inwards due the two Torx screws holding it not being very tight. The magsafe connector seems to connect (magnetic), but the full electric connection actually got obstructed by the outside of the lower shell of the Macbook. Due to this the pins didn't all connect correctly. I guess the middle pin of the connector simply didn't link. (I was able to boot without battery remember, so two outer pins must have connected alright)
Shifting the Magsafe port connector further out and retightening the screws (one of them below the lcd data cable connector) solved it for me.