I have a similar issue. I just installed a new alternator, and car starts right up. Measured voltage with engine off (after charging a few minutes) and battery 12.28 V. Then tested with engine running: 13.63V. Drove around the block and yet- my ignition check light stays on all the time.
Small wire is connected to alternator. Since it is a new alternator, I'm confident it's something else. I had the same problem on my old alternator and then it stopped working and would not charge the battery when car running.
I guess my question is - does the wire from the alternator go straight to the bulb, then out the other end to the ignition switch? Wiring diagram looks that way. I can then follow the wire testing for continuity. Would that help?