First of all, you should distinguish two concepts. Sigma and π refer to bonding modes, and 2p is the name of the orbit. This cannot be confused. Sima is a bond that must be formed, and the bond that meets the head is the strongest, so it is formed first. If it is a sigma formed in the direction of 2px, and both atoms have lone electrons in the direction of Y, then it will be π. Because the X direction has touched the head, it is impossible for the Y (or Z) direction to touch the head, and it can only be π side by side.

You can understand that the x\y\z in 2p refers to the direction of the x\y\z axis of the space coordinate, and all your previous understandings are correct.

If you are a high school student, I suggest you don't dwell too much on the book material structure. Many of these contents are guesses in themselves, and some things even chemists can't explain. You just need to master the rules and solve the problem. For example, the single bond must be suitable, the double bond is suitable and π, and the triple bond is suitable and π.