'02 Daewoo Nubira 1.6 car battery and handbrake light lit at the same time

Hello,
recently I've seen a couple of times after starting car (hot or cold) that my car battery light (and handbrake light) are just barely lit, than lights up more bright or dimmer after a while...
If I press on the gas pedal, it will disappear seeming like everything is normal. It doesnt always happen, just a few times in last 5-6 days.

Also, in one instance (while they were lit), my car died on me while driving (it has gone to idle in between gear changes - manual transmission), I immediately fired it up again and it fired normally, didnt happen again.

I've tried refilling car battery after that, but it was almost full anyways, voltage was 12.9V approx, it filled to 13.1 and stopped filling in matter of minutes.

Yesterday I've measured voltage on the battery while car was running on idle in different conditions:
under no load (lights were off, radio off, heating off, everything but the engine off): - 13.5 to 14.1V if I remember correctly (should've written it down)
under middle load (lights were on, radio on, low volume, heating on to number 3 setting): - 13.2 V
under heavy load (lights, both short and long on, radio on full volume, cabin lights on, fog lights on, heating to full, turning lights on, rear window heating on...) - it eventually dropped to 12.1 V

Remember, my car battery with car off measures from 12.7-13.1 V!

Also under heavy load and at 12.1V measured, there was still no car battery or handbrake light on on the dashboard.
But I've noticed red battery cable contacts were hot (I couldn hold my hand on it longer than a second or two), black contacts were cold...

When my wife pressed on tha gas pedal under full load to around 2000RPM, voltage went up to about 13.2V.

Is my alternator dying or what?
Are those readings normal or not, does the fact that the red contacts are hot means that car is pulling power from a battery instead of alternator (generator)?

If my generator is dying, what would be the first suspect? Diode bridge or what?

Also, I've got Hantek1008c and would like to know how to pull additional measurements on the generator to know for sure if it needs refurbishing.

Thanks in advance guys!!

update:


Is this behavior normal or not?
While rotating rotor by hand, sometimes it lights up dummy light, sometimes it doesn't, also, on some positions of rotor small push on the rotor inwards turns off/on light, and on some positions of rotor small pull on the rotor outwards turns off/on the light.

Thanks in advance to any comment.

It was bad alternator, changed it and its good now.