I also measured the voltage at the thick wire on the alternator. It was within .25 volts of the battery voltage, around 12.30 or so.

That 12.30 was the battery voltage. The thick red wire is directly connected to the battery positive (by way of the Starter main terminal). So you should always read battery voltage there, or close to it, as you indicate. That's what you'll see with a dead alternator.

When you have a working alternator, its output (say a nominal 14 volts) it will show a the the alternator and at the battery positive as well—because those two points are electrically the same. When you measure round 12.5 volts at either end, it's from the battery. Over 13.5 volts would be the alternator output, wherever you read it.
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Bruce Young
'93 940-NA (current), 240s (one V8), 140s, 122s, since '63.