First Arduino electronic project

Hello!
I'm doing my first Arduino/electronic project and I'm having some troubles understanding how to do wiring and how to handle power supply

First of all I have an Arduino UNO R4 Minima, and my goal is to do a simple mini game with a led strip and arcade button

Here is what I bought for the project

I thought I was gonna power the LED strip with the power supply using the GND and 5V cables, that would go through 2 WAGO to also power the Arduino with the same power supply.

The issue is that I do not understand how the cables from my LED strip are working
I have 2 loose cable (GND/5V) and a 4 pins female adapter that do GND BO DO 5V:

Do I need to buy an adapter for this? Since my goal is to control the LEDS using the Arduino
Thanks for your help!

They are actually BI and DI but labels are covered by heatshrink.
You can use the "loose" cables and wagos to connect PSU 5V/GND to both strip and arduino.
Then use jumper wire to connect DI to your arduino digital pin.

Thanks for your answer!
Just to be sure I understood
5V GND (white/red "loose" cables) connected to power supply
Then out of the 4 pins adapter -> green cable (DI) with a dupont to the Arduino (no need for resistance?)
Btw thanks for the explanation I didnt understand why it was DO/BO first and not DI/BI!

Yes. On the "DI" (data) line.

Yes. Add an electrolytic capacitor to help with power fluctuations during "bright" colors. The values I have seen for the capacitor range from 100uF to 1000uF. I have used both and never saw a difference.

Here is a generic drawing for powering a WS2812.

Just to clarify, the B stands for Backup and the D stands for Data, I stands for Input, O stands for Output.
If there is no meaningful signal on the DI, the WS2813 will choose BI as it's input signal. DO is the main output, basically the output of the current pixel, BO is the backup output and is a direct forward of the DI. If the pixel somehow breaks there will still be the incoming circuit being forwarded. These strips are great for situations where there is no easy access to the strip once mounted.

Correct. Since you use same power supply for both, you only need to wire DI from the 4-pin connector.
Resistor is not strictly needed, but it can improve signal in certain situations and gives extra protection.