Yes, that funny little plug is for the PS. Its part of the speed sensitive steering BS. Mine broke off and I never looked back.
Get the laternaotr tested or just get another one. The problem is, the rule of thumb for alternators is you will have to go through about 3-5 of them before you get one that lasts. They come with a warranty for a reason. They suck... Same goes for starters. But sometimes you get lucky.
Have fun.