Loco is right. It could be BOTH!
Here's a little insight from years of troubleshooting automotive charging systems. It seems long winded but it's full of some good info.
When the engine is running, put a volt meter on the battery terminals. It should read something ABOVE 12 volts. If it does, turn on the headlights and watch what the voltage does. If it drops a bit then comes back up and stabilizes ABOVE 12 volts, alternater function is good. If it drops and stays dropped, and or, keeps dropping, you have a weak alternator and maybe a bad battery.
Here's the deal with alternators:
They need voltage to energise the electromagnet (field winding) inside them. The part that spins, the rotor, has poles that pass by the poles in the field winding and this produces pulses of AC electricity. The diodes inside the regulator, usually built into the alt., smoth out the AC to DC voltage. This then flows through the battery. The regulator determines how much flow based on the current draw on the battery.
When you do the "pull a battery cable while it's running..." test, there is enough residual magnetism in the field windings to make enough voltage to keep things going. If it stays running, you know that its making voltage, but but it may not be making much current.
If your battery is borderline, it might test OK. Did they do a FULL LOAD TEST???? It might fail that test. THAT'S why it starts when you add another good battery while jump starting it.
A note on batteries is in order here, bear with me...
When you connect two batteries together as in jump starting, they will equalize towards the lower voltage battery. This is why a car with a completely dead battery needs to sit and charge before it will even crank over. In fact, it will start right up if you just bypass the battery and jump it. The dead battery is an extra load on the charging systems peak voltage output. If your battery has a dead or weak cell, it would not have enough amperage to operate your starter by itself. Jumping it does.
First things first. Get the battery tested at full load. I would suspect your battery has a bad cell based on what you've said. A new one will run you less than $80. Well worth the bucks. A good battery is EVERYTHING in any car. A must have. Once you've verified the battery being OK with a full load test AND a hydrometer test on EACH CELL, then you should get the alnernator tested at an auto parts store. Most will come out to the parking lot and test it with a hand held unit or a roll around, cart mounted older unit. ...and they'll do it for free.
Full load test, Full load test, Full load test, Full load test, Full load test....
one more mechanics opinion.
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