I think in practice the simplest approximation is probably always good enough. The reason is that I don’t think we care about collision probabilities larger than 50%, and the value of k that gives p=0.5 is sqrt(n) according to the simplest approximation (p=k^2/2n), while p=0.39 for k=sqrt(n) according to the exponential approximation (p=1-(e^-(k^2/2n)). The difference at the most extreme value we care about (p=0.5) is small enough (~11%) that I think the simplest approximation should always suffice for practical applications.