You really don't want to disconnect the positive side like that on modern cars. But check the clip on the side. The alternator only puts out the voltage/amperage it needs to keep the battery charged and it uses a wire on that clip to tell it what to produce. Its been a while since I worked on an alternator but I can look for it if you need it. One wire sends the signal to the cluster that is meant by the same voltage coming from the ignition switch and this is what controls the battery light. The other wire is the exciter/load signal wire. You can also undo that clip and see what the voltage output is. Some alternators require that it get a signal from this on start up to tell the alternator to work so I'd unplug it after you start the car. Then check as output should raise if this wire is the problem. You can rewire this really easy if the factory wire has gone bad.
Also if those wires are clean and have good contacts the other usual cause for low voltage is a bad diode. But since you switched alternators I would doubt that is your problem.
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