To answer this question specifically, Fatou has essentially two different plotting modes controlled by the iter boolean keyword. In the example you referenced, the coloring function e^{-|z|}\cdot n^p is used with the limit values of z, which causes a coloring value that also varies on the inside of the set.
However, if you want to get the traditional iteration count display of the Mandelbrot set, then you need to set the iter keyword to true as such
mandelbrot(:(z^2+c),n=700,N=20,∂=[-1.91,0.51,-1.21,1.21],iter=true,cmap="gist_earth") |> fatou |> plot
Then the coloring scheme works as you typically expect it. The title of the plot tells you what kind it is.