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Just FYI, don't pull a battery cable while engine is running, you can fry electrical parts doing that.
The voltage with engine off and on tells you the alternator is pulling power from the battery instead of charging the battery.
So yes, engine spark(coil) is running off the battery so when battery is disconnected engine dies.
From your first post you have a 1994 explorer alternator with wiring harness.
You have to have the extra wires hooked up to the alternator or you will get no voltage out of the alternator, they run the voltage regulator.
There should be 3 wires besides the larger wire hooked up to the alternator, one of those is a short jumper wire.
"S", "A" and "I" are the wire labels.
"S" is a short jumper wire
"A" runs to power box or solenoid
"I" runs to the charge gauge or idiot light in the dash board, then to ignition and other systems.
"i" won't have voltage unless the alternator is outputting voltage
"s" is the jumper wire to control alternator voltage so needs to be connected back to another alternator connection.
There should be a 4th(5th counting b+) connector, with a spade connection that the "s" wire plugs into.
This diagram might be easier to understand:
You should have the 2g setup
http://www.grandmarq.net/vb/showthread.php?29841-alternator-wiring-diagrams-1g-2g-and-3g
the white/black wire, "s" to "s" jumper, is what seems to be missing that is what allows the alternator to output voltage.