Battery or connections. Alternator seems normal from what you describe, factoring in the gauge/needle tolerance and voltage drop. Easiest thing to do is have battery checked, but it is VERY hard to find someone who can do it right. Flaps people may tell you it's OK when it's not. Their little hand held tester in the hands of a person who forgot the training gave me fits. I just kept going to different flaps til I found someone who knew what they were doing. It was a bad battery, it always is.
My opinion comes from :
1. Alternator shows 14V after start, normal. Seems to settle down after a while. Normal.
2. Alternator trying to keep battery "topped up" during driving because regulator sees low volts...... but 12.16V after getting home. NOT normal. So guessing battery. But another warning, bad cell and etc. needs a person to LOAD TEST with a good tester and know-how to use it. I think they hold the load for 15 seconds.
3. Of course 12.16V can be caused by bad alternator also. But rebuilt you say. So battery! sulfation, bad cell, hard winter ....or CONNECTIONS. Corrosion, loose, broken cable.....
If you have done anything recently let us know. Wiring, new parts, noises under hood, age of battery. Just be sure that fan/relay is not doing some over-draw or intermittent shorting out. You could disconnect and test drive maybe.
