Your power supplies (except in quite unusual situations beyond Arduino) should all have their 0V / GND wires tied together - no other topology makes sense.
This basic understanding of voltages (potential difference) is critical to wiring up any electrical circuit.
GROUND is a potential that is nominally equal to the surrounding electrical ‘environment’.
POSITIVE voltages are those ‘above’ ground. +12V, +5V and +3V3 are the common positive voltages.
NEGATIVE potentials are those voltages below ground potential.
In virtually every ‘standard’ Arduino project, there are no negative voltages exposed to the developer/user.
Custom boards and interfaces may add ‘other’ voltages needed by the application.
So the take-away is : BEFORE you burn out boards, or start programming, make sure you have your hardware worked out. There’s no point putting fuel in your car if it doesn’t have tyres!
Number 3 is not a type of ground but a type of connector.
Number 2 is any point you decide to be ground is ground, it is no different from any other sort of ground apart from mains ground which is soil or dirt under your feet.
LMI1:
Weirdest things I have seen happen when someone forgets to connect grounds together.
I managed to do this, leading to much head scratching, replacing the starting motor on a '72 Cadillac with a 500cid (8.2L) engine.
Gosh, it spins, but not hard enough . . . somehow, I managed to not attach an inch wide flat woven connecter that (for good reason) goes from the negative battery terminal to the chassis on a starter bolt . . .
It drove home ground issues in a way no hobby circuit ever could have managed!
lastchancename:
Don't worry, smoke works just as well in cars as it does electronics
It wouldn't be the first car I let the magic smoke out of . . . but in this case, it merely meant that not enough current got delivered from the battery's local connection to group, and the starter just spun weakly.
As for limp modes . . . I had a truly interesting drive home in an '02 Deville DHS the day I bought it. The prior owner having set all kinds of stuff putting in the wrong plugs, it did 0-30 in, roughly, 30 seconds.
Yet it could reach freeway speed (albeit slowly).
New ignition banks for each side (I forget what they're called) and the right plugs, along with a couple hours of my time, and all was good . . .
ballscrewbob:
10 minutes and no charge and I was on my way both grinning and feeling sheepish.
That is very familiar feeling. I once? searched a fault on mains powered machine. I started from the sensors, very methodically I went through whole machine until I noticed the mains lead was unplugged.