Investigation of the triggering mechanisms of radio active galactic nuclei (radio AGN) is important for improving our general understanding of galaxy evolution. In the first paper in this series, detailed morphological analysis of high-excitation radio galaxies (HERGs) with intermediate radio powers suggested that the importance of triggering via galaxy mergers and interactions increases strongly with AGN radio power and weakly with optical emission-line luminosity. Here, we use an online classification interface to expand our morphological analysis to a much larger sample of 155 active galaxies (3CR radio galaxies, radio-intermediate HERGs, and Type 2 quasars) that covers a broad range in both 1.4 GHz radio power and [O III] λ5007 emission-line luminosity. All active galaxy samples are found to exhibit excesses in their rates of morphological disturbance relative to 378 stellar-mass- and redshift-matched non-active control galaxies classified randomly and blindly alongside them. These excesses are highest for the 3CR HERGs (4.7σ) and Type 2 quasar hosts (3.9σ), supporting the idea that galaxy mergers provide the dominant triggering mechanism for these subgroups. When the full active galaxy sample is considered, there is clear evidence to suggest that the enhancement in the rate of disturbance relative to the controls increases strongly with [O III] λ5007 emission-line luminosity but not with 1.4 GHz radio power. Evidence that the dominant AGN host types change from early-type galaxies at high radio powers to late-type galaxies at low radio powers is also found, suggesting that triggering by secular, disc-based processes holds more importance for lower-power radio AGN.