If I do a SMC reset, then the battery will charge.
I am having a very similar problem.
If I do a SMC reset, then the battery will charge.
If I don't do a reset and the computer gets turned off or the adapter comes out of the MagSafe, then it will not charge.
The computer will run on battery only as long as it has power remaining.
If either of us figure out the problem, we should tell the other.
Is this a good question?