Can you really pull a battery cable and the engine still runs?

and the voltage source for the field coil comes from the battery itself.
That's where your veering off course.
Power comes from the alternator, as long as its the highest voltage power source.

That's was a quick way to check if your alt was charging, pull hot cable off battery, if still runs alt is charging.
For the poor guys with brand X that didn't have ammeter. Otehrwise just look at the ammeter. If the battery is discharging then the alternator isn't supply power, or enough power.

If your battery is dead and you jump your battery to start the car then disconnect the second battery isn’t the car basically running on the alternator until charged? Which is common?
Correct. And its still running on alternator after the battery is charged. That's why the ammeter shows no current flowing out of the battery.
If that were totally true, pulling battery terminals off and back on wouldn't be an issue....but it is. The voltage spikes can pop diodes. The regulator won't prevent that. A battery in the system does.
Yes.
Lets explain this a little more if we can.
The voltage regulator controls current flowing to the windings that create the electromagnetic field. When the regulator cuts current, the electromagnetic field begins to break down. When current is allowed to flow, the electro-magnetic field builds up.

The alternator output on these older cars and trucks is only dampened with a capacitor. The battery connection is on branch at the main splice, so it helps dampen some noise, but not as much as on systems where the alternator connects more directly to the battery.