How to empty cart or clear session "externally"

This one should work as it removes all items from the current quote:

Mage::getSingleton('checkout/cart')->truncate();

But you need to save it afterwards:

Mage::getSingleton('checkout/cart')->save();

By the way, you can remove the following line:

$resource = Mage::getSingleton('core/resource');

since you don't use the variable and instantiating this singleton doesn't do anything in particular.