It’s the voltage regulator and it is almost certainly part of the alternator. Best option is to replace the alternator. I once had a weird problem where my rebuilt alternator would shut-off when it got hot. So if I took it to get a warranty replacement, they would test it, and it would look fine (since it wouldn’t be hot). Drive it for 15 minutes, it cuts out - but they can’t do that on a test stand. So, I figured the best option was to hard-fail it. So I plugged in 120V to the voltage regulator terminals (the smaller wires) for a couple of seconds and nothing happened. Then I ran the alternator on the car and the voltage immediately zoomed up to 16V (just like you)...I had popped the voltage regulator electronics and it no longer regulated. At that point, I took to to the auto store and got my warranty replacement.
In your case, you’re killing your battery with such a high voltage, and also stressing all the electronics in the vehicle - get the new alternator and be done with it.
You’re probably right.
But the thing is I can’t figure out why, with the new battery, the first time I started it the light came on almost immediately - and the voltage tested 16v.
But EVERY TIME I started it since then, it seems to take more and more for the light to come back on. I just drove it to the mailbox and it didn’t come on!