What determines the coefficients of the laplacian filter?

Question

I am building an approximately isotropic Laplace kernel based on the guidance in enter link description here. What I don't understand is the derivation process from (2) to (3). I don't know why the coefficients of the two kernels have changed from 1/S² and 1/2S² to ω and (1-ω). I mean, since we already know that the distance from the center to the top, bottom, left, and right is 1, which is a pixel size, and the distance from the center to the corner is the square root of 2, isn't the ratio between the values at the corner and the values at the top, bottom, left, and right determined? Why do we still need to provide an ω for us to change? Why do we want the sum of these two coefficients to be 1?