Some vaccinated people still get severe disease, and others even die (both at significantly lower rates than for the unvaccinated, of course). The reasons why this happens in some cases and not others is uncertain, and, accepting the Bayesian interpretation of probability as I do, I would describe that as reducing the probability of disease (and also death). Put differently, you cannot tell a priori whether being vaccinated will definitely stop you dying, but you can observe that the a posteriori population level rate is reduced. You are correct that it also reduces the probability of transmission.
Vaccines reduce the severity of disease, sometimes to the point of no noticeable effect. They may reduce probability of transmission.