12v LED strip + 5v arduino- how to direct power

The LED strip ( https://www.amazon.com/gp/product/B01CNL6LLA/ref=oh_aui_detailpage_o02_s00?ie=UTF8&psc=1 ) has 5 inputs:

{
Power
Ground
Data
}
Power
Ground

I assume the way this is supposed to work is that the arduino provides the whole 'group' (power/ground/data) via its corresponding pins, and then a 12v power supply provides the remaining two to power the LEDS.

In this way, there's a 5v signal being sent as power/data to the little chips, and then 12v power being supplied to power the LEDs.

Is this accurate?

If not, what's the point of the two power/ground wires? And how else ought I configure the 12v vs 5v difference?

I assume the way this is supposed to work is that the arduino provides the whole 'group' (power/ground/data) via its corresponding pins, and then a 12v power supply provides the remaining two to power the LEDS.

Not sure what you mean by whole group, but basically you need 12V power for the lights and you need to get that down to 5V for the Arduino. If you are running little else you can use the onboard regulator to get this.

The point about the two power connectors is to reduce the effective resistance of the power strip by wiring both ends to the power and ground using thicker wire.

gah shoot I think I fried my board...

So, I thought this was how it worked:

12v wall wart -> arduino 5v -> power
ground -> ground
pin N -> data

different 12v wall wart -> power
-> ground

and then connect the "ground"s (for reasons I don't fully understand...)

But, if in reality, both power's and both ground's are the same, what I've likely accidentally done is connected a 12v power supply directly into the 5v pin of the arduino. (I had assumed there were two electrically isolated systems- one for communication and one for power...)

and now, the arduino isn't being recognized as an available usb port.

So, have I fried my board?

Also, to clarify, the reality of the situation should be: (please correct me if I'm wrong)

12v wall wart -> arduino 5v -> leave open
ground -> ground
pin N -> data

different 12v wall wart -> power
-> ground

and then, do both grounds need to be connected? or is the fact that the arduino is connected to one ground lead and the wall wart is connected to the other enough (if they are in fact the same ground, they're by definition connected... right?)?

and then connect the "ground"s (for reasons I don't fully understand...)

Have a read of this:-
http://www.thebox.myzen.co.uk/Tutorial/Power_Supplies.html

what I've likely accidentally done is connected a 12v power supply directly into the 5v pin of the arduino.

Yes that could kill things, most likely the USB / Serial converter chip on the Arduino.

Also, to clarify, the reality of the situation should be: (please correct me if I'm wrong)

12v wall wart -> arduino 5v -> leave open
ground -> ground
pin N -> data

different 12v wall wart -> power
-> ground

and then, do both grounds need to be connected?

Yes both grounds need connecting for it to work. While this will work you don't actually need two wall warts.

12v wall wart -> power jack
pin N -> data
same 12v wall wart -> LED power
-> LED ground to Arduino ground