We investigate the cutoff phenomenon for Markov processes under information divergences such as $f$-divergences and Rényi divergences. We classify most common divergences into four types, namely $L^2$-type, $\mathrm{TV}$-type, separation-type and $\mathrm{KL}$ divergence, in which we prove that the cutoff phenomenon are equivalent and relate the cutoff time and window among members within each type. To justify that this classification is natural, we provide examples in which the family of Markov processes exhibit cutoff in one type but not in another. We also establish new product conditions in these settings for the processes to exhibit cutoff, along with new results in non-reversible or non-normal situations. The proofs rely on a functional analytic approach towards cutoff.