I think it's the alternator. Once the engine is running, the battery really isn't used anymore. The alternator supplies the voltage that the vehicle needs to run. If it stumbled after disconnecting the jumper cables, then you were running of your friends alternator, and when you put the load on yours, it failed.
Try this. Get a flathead screwdriver, and with the engine running, carefully set it against the back of the alternator case (be careful not to touch any wires or connectors). If you alternator is working, you will be able to let go of the screwdriver, and it will not move. A properly functioning alternator creates a magnetic field, which will hold the screwdriver to the case. If it falls, there is not a sufficent charge being generated to create the magnetic field.
Take care,
~Chris
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