no you cannot kill your alternator by switching your battery from one car to another
The alternator/generator
The alternator or a poor wiring connection.
test the voltage on the battery first, then on the alternator. If its good then it could just be the diode in the alternator. I would suggest you replace it becuase eventually your alternator will die and the battery will not get charged leaving you stranded somewhere. It is possible to just be a battery but you wont know unless you have the battery tested. check the voltage on the battery and the alternator before you do anything else.
altenator
Check for blown fuse(s) or blown fusible link(s).
Its probably not the battery, its more than likely the alternator that isn't keeping the charge, to check it turn your car on and the unplug the battery, if it stays on then its something else, but if it turns off then get a new alternator
Battery light usually means alternator not charging. Most common solutions would be be a bad alternator, bad connection, or loose alternator belt. If the battery and alt test ok than check the connections to the alt. You may also have a battery drain (something that is staying on and killing the battery)
There isn't a black connection on the alternator? That would be the correct side. If not, you could almost connect it to the bolt that holds down the alternator to the frame of the car. Check with someone else to be sure, but the negative side or connection is the common or ground wire. Just avoid hooking it to red.
if both were replaced and installed properly you may have an open circuit.a test to find out where the open circuit is , you need to placed a multimeter on your battery and have someone pull out each fuse one at a time until you see any change on the voltage reading of your meter.(hopefully someone else has an easier test for you.)
wires, connecters, terminals
There is a problem with your charging system. May be a defective alternator or something else.
Disconnect the cable from the alternator and see if that stops the draw. If it does, the alternator may be defective. Otherwise there may be something else drawing current.