There are a few notebook power supplies I know of where you can hit a button on the brick and stop charging the battery - though I've only seen in on HP business models.
What do you mean the battery isn't being charged? Does the system tray report that the machine is on AC power?
'power cord' is just a common term; it doesn't really apply to any specific piece of hardware (other than a cable that supplied power, which can be a lot of things).
'battery charger' usually describes a piece of equipment dedicated to charging a battery (and not the rest of the computer) and generally they're not available for notebooks.
Though, through consumerism, lots of words get tossed around in places they don't belong.