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I'm the developer of DHARMa. First of all: note that results are not actually conflicting - a non-significant test doesn't mean that there is no overdispersion, it just means just that the respective test is not sure there is. Specifically, DHARMa and overdisp or the newer performance::check_overdispersion perform different tests, and what you see is that ...

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Implicitly, you are suggesting that there is one effect of your predictors on your responses, and that your responses really are "only" three different ways to measure this effect. If you want to throw them together, and if they additionally seem to derive from different distributions, I can only think of a hierarchical (Bayesian) approach: Option 1: model ...

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take-home messages The negative binomial uses more information from the data, so it's expected that it would be slightly more powerful than the rank-based Kruskal-Wallis test. In general, you'd use K-W if you were concerned that the distributional assumptions of your model were badly violated. The difference between $p=0.021$ (overall significance of the ...

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