Is it an alternator problem?

Hi, over last few months my car has the following
symptoms:
1. Car starts then die when the headlight turns on.
2. Car is driving slowly (with headlight turns on) then die
when high beam turns on
3. Car is driving with speed (40km/h), suddenly die then
back up immediately.
4. Car suddenly die when turn (at low speed).
5. I use a multimeter and check the voltage at the
battery terminals everytime the car starts. In one
instance I see the voltage reads at 12.8 (it normally
reads 14.0 or 14.5 when car starts). Turn off the engine
and start again it reads back to 14 range.
6. I check the belt connect the alternator and it spins the
alternator fine.
I suspect it's the alternator. However all these issues are
intermittent, 80% of time the car runs ok. What kind of
tests should I do to make sure it's the alternator fault?
Also I read somewhere that's it's likely just the brush
failing so I only need to replace the brush, is it true?

Thank you,

P/S: when I said car die I meant all of sudden electric
power is gone, the car stalls.