In the past papers, when we need to test the goodness of fit, how can we distinguish from the question if we need to use E*lambda/sqrt(E*lambda) (i.e. Poisson approximation) or E*q/sqrt(E*q*(1-q)) (i.e. Binomial approximation)?
You should look at the parameter to understand if it is a Poisson parameter i.e. if it is a rate vs. binomial where it is a q type probability.