The current statistics around breakthrough cases are not very clear (we have case studies but are lacking broader studies). We do know that vaccinated people can get the virus, can develop an infection, and spread it. We do not yet know exactly how frequently each of those outcomes occur, but the data seem to show that the possibility of each is definitely lower for each of the steps in spreading it. So until more data arrive to dispute this point, to the best of knowledge vaccinated people spread the virus less by virtue of: reduced ability to contract the infection and reduced time of being able to spread if infected. I don't know how much reduction is offered, but the post to which you are replying is not rendered inaccurate by vaccinated people having the capacity to spread the virus.