Show/hide and move points using random

Hi all! I’m trying to create a system that can appear and move points 2 randomly within given boundary. I want to make random controllable and don’t want points to be appearing at the same position. So far I’m having trouble moving the points in 2 dimension and it’s always moving 3 dimensional in positive direction. Also, I want to combine this moving function with random appear. How would I do this?
Drawings I attached are the designs I’m trying to achieve using this system.

dot (17.8 KB)

Something like this?

dot (15.4 KB)


YES!!! Wow you are genius!!
Thank you so much!!!

Check this as well.

dot (16.3 KB)

Oh hahahaha
I was trying to solve that. haha
Amazing!! Thank you so much!!
I also want to make it clear about the script that you wrote. Basically the top part of the script will be the shape that will be identified and bottom part will be random patterns? If thats the case, if I want to do layered rotating cross patterns or rotating triangular patterns my script will be more similar to the bubble pattern?