Power Source status not changing from Power Adapter to Battery

I have the late 2008 aluminium unibody 2.0 GHz MacBook. When I disconnect the power adapter from the MacBook, then the Power Source in the status bar still shows that the Power Adapter is connected, even though it isn't.

When I connect the Power Adapter, the Power Status in the status bar changes to "Charging", and the battery charges correctly. If I then disconnect the Power Adapter then the status of the power changes to "Not charging" and still shows that the Power Source is the Power Adapter. In the System Profiler under the Power section, the AC Adapter Information also shows that the Power Adapter is still connected.

Usually when I disconnected the Power Adapter, it use to show that the Power Source is now running from the "Battery". This is not the case anymore, as the MacBook seems to "think" that it is still running from the Power Adapter (even though it isn't).

Consequently, I don't get the Low Battery warning anymore. While working on the MacBook the battery will drain to 0%, and then just suddenly it will terminate without a warning. Usually it would have first warn me that the battery charge levels is low, and then after a while it will go into hibernation mode.

Things I already tried, without any luck:

1. I tried resetting the PRAM as well as the SMC.

2. I bought a new battery. With both the old and new battery the issue persists.

3. I tried 3 different AC Adapters. Two of which is 60W adapters and the other one is a 85W.

4. I discharged and recharged the batteries.

5. I reinstalled Mac OS X.

Does anyone maybe have an idea how I can get the MacBook to recognize again that it is running from a "Battery" as Power Source when I disconnect the Power Adapter?