Using a different permutation at each step leads to a simulation with more isotropy and without cycling behaviour.
Every cell (of N) is updated exactly onece per time step.
at each time step, each cell has probability $p$ of possibly updating its state.
I have seen asynchronous models where there is a single ordered list of the individual cells with each cell firing one after the other, with the same ordering maintained over multiple cycles, or with a different ordering being generated each time after all of the cells have fired.
Some of the schema you listed and some others are at
I am interested in asynchronous automata, where they do not all have to update simultaneously.