sounds like you figured it out. it's what I personally thought it was gonna be after you said there was basically nothing else connected except stuff to make it run. otherwise I figured a stuck relay that won't turn off. again, a pic would be great just so we can see the style of alternator you are workng with. do you have another spare one you could lay next to the other one and simply do the wiring on it without mounting it? use a jumper wire to ground the housing. would be worth doing just to see if it is your wiring or the alternator that is giving you an issue.
this link explains it pretty well. basically the one connector in the rectangular push in plug (the one closest to the battery cable connection charge wire) is not really needed unless you want a "no charge indicator" light on the dash.kinda a good idea I think but thats just me. this works by feeding keyed battery power to the light on the dash. the other side of that light connection that usually would go to ground actually goes to that terminal on the alsternator and uses that connection as a ground when the alternator isn't charging. that makes the bulb light up according to the amount of ground the alternator is giving. this may be dim or maybe even pulsating when the engine is first started and hasn't been revved up yet to get the alternator to charge. when the alternator has been revved up some it will start charging and feed power down the ine that is connected to the charge indicator bulb. this effectively gives the bulb power on both sides of the bulb filament so it will go out because it just lost it's ground circuit. the other terminal on the plug in connection, the one that goes to the ignition switch and is powered with the key on, is the one that will "excite" the alternator and make it charge. a lot of guys wire that directly to the battery charge connection on the alternator. some of these alternators have a third terminal in the plug whicch is for running a tachometer. not too many of those out there so you don't likely need to worry about that.
https://smithcoelectric.com/blogs/te...wiring-diagram
and here is another link that explains it further in depth than what I could muster up from memory. haha
https://vintageautogarage.com/p10si-...ug-with-diode/