Laplacian Operator with and without Diagonal Direction Elements in the Kernel
All of those are Discrete Approximation of the operator - Laplace Operator.
You can discretize it in any logical manner.
In the cases above, Istotropic means if you rotate it it looks the same.
The diagonal elements means the filter would be sensitive to changes which are in 45, 135, 225 and 315 degrees, it it means it will also amplify more noise.