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The easy thing to do first is check & clean the connections between the Alternator, Starter Solenoid all the way back to the Battery. If that looks good use a milt-meter (volt/ohm-meter) and measure the resistance and/or the voltage between those locations.
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Correct; the ignition light only glows when there is a voltage differential between the battery and the charging circuit. The differential can be due to either a bad alternator, a bad battery, corroded connections or a combination thereof.
First thing to do is clean ALL of the connections: battery cables, posts, ground cable where it connects to the chassis, the alternator lug at the solenoid, etc.
Are you sure the battery is holding a charge? If you have +14.5v at the alternator, but only 12 or so at the battery, I'd suspect a shorted battery cell or high resistance at the connections/cable. The opposite suggests the alternator.
Does the light dissapear when you rev the engine?