(b) That's right. It may seem that B is only promising to pay what she already owes A, making this a case of 'past' consideration. But in fact this is a special type of case. The parties genuinely disagree about what is owed. A claims more than B admits she is liable for. B then promises to pay something more than she admits is owed, and A compromises by giving up the rest of his claim. In cases such as this, the compromise is a sufficient benefit and/or detriment to be good consideration, and the agreement is enforceable.
However the compromise must be based on a genuine belief (even if possibly an incorrect one) in a valid legal claim or defence. In the absence of such genuine belief, any compromise is illusory and does not provide consideration.