The test by removing the battery cable and the car stalling is not recommended on modern cars, and can actually cause damage to the electronics. So I'd say that test is inconclusive. The battery acts as a large capacitor and filters a lot of the alternator's noise before it gets on the electrical system. The alternator produces AC, which is then rectified and filtered by the battery to provide a DC signal to the car. When you open the circuit to the battery, the unfiltered noisy rectified signal makes it to the PCM and other electronics which may react unpredictably.
Do you have a voltmeter to check what voltage is on the battery when the engine isn't running and also when the engine is running? If not, you can pick up inexpensive meters at many of the department stores, hardware stores, of if you're really looking for an inexpensive one, Harbor Freight has them. To do much more troubleshooting of the charging system you're going to need a meter.
I trust you've also double-checked the connections at the alternator. You'll also want to check fuse 26 (30A) in the Engine Compartment Fuse/Relay Box.
-Rod