I have seen in some places that the model often named in the literature as zero-inflated beta model is actually a hurdle model. My question: is the zero_inflated_beta() model implemented in brms a hurdle/two-parts model? If not, what is the difference?
Yes, most parameterisations of a zero-augmented (aka zero-inflated) regression model and/or zero-and-one augmented are a hurdle-type model. See for instance:
Liu F, Eugenio E (2018) A review and comparison of Bayesian and likelihood-based inferences in Beta regression and zero-or-one-inflated Beta regression. Statistical Methods in Medical Research 27: 1024-1044
Wright W, Irvine K, Warren J, Barnett J (2017) Statistical design and analysis for plant cover studies with multiple sources of observation errors. Methods in Ecology and Evolution 8(12): 1832–1841