Powering Nano + LED strip w/ 12 volts from single outlet

I asked a question a bit ago about using a single 12v 10a power brick to simultaneously power both an Arduino UNO (R3) and a WS2811 LED strip (also takes 12v).

What I settled on (and appears to work!) is the following:
I connect the arduino power/ground in parallel w/ the LED strip- both directly from the power brick. The arduino takes 12v into the VIN pin, and ground into the GND pin. The LED strip takes 12v into its power line, and connects its ground to arduino GND. (There is also a data pin that connects the arduino + strip, as the arduino is driving the LEDs, which justifies why the arduino and the strip share the same ground [at least, this is what I was told to do!]).

So, that setup works (or at least, it hasn't burned anything down as of yet). But I was also told that applying 12v directly to the arduino VIN was feasible only w/ certain arduinos (with a confirmation that the UNO was one where it does work).

I'm looking to port the project over to a Nano (reduce spacial footprint, + cost, + easy solder points), and expect I won't be able to just power it via 12v to VIN.

Is there an established process for this?

Here's one proposal that I would love if you all could sanity check:
I bought one of these ( https://www.amazon.com/gp/product/B0758ZTS61/ref=oh_aui_detailpage_o01_s00?ie=UTF8&psc=1 ). So I could power the strip in the same way that I currently do, but rather than just plugging the 12v into the arduino VIN, I instead put this regulator in between (bring the 12v down to 5v) and then plug its output into the arduino 5v.

This seems fool-proof, but I'm skeptical because I don't fully understand the circuitry behind how I'd combine the strip ground + arduino ground (which apparently do need to be combined for the data line to function).

What seems odd to me about this is that the ground that connects to the strip is used in a 12v circuit, and the ground that goes to the arduino is used in a 5v circuit. I feel like I'm missing a fundamental understanding of electronics here... Should I put a diode on the arduino GND to make sure that 12v doesn't get pushed into it?

Any clarification that would help me get my head around this would be very helpful! Thanks!

No diode necessary. Your drawing is right. The 12V ground and the 5V ground from the regulator are the same thing, they are connected on the regulator board. Take your DMM and measure the resistance between the 12V ground terminal and the 5V terminal of the regulator board to see.

No, don't apply that diode. GND from the 12 volt source is the same GND as the 5 volt regulator uses.