A good summary, but there is a little more nuance related to Cardinals. They were originally the principal clergy of Rome, back 1000 years ago. They have gone through many changes over the years, but always retain that association with Rome. New cardinals are assigned to parishes in Rome (or dioceses in the province of Rome.)
With the codification of Canon Law 100 years ago, only priests could be named as cardinals. For different reasons, St John XXIII made it a policy that only bishops would be cardinals. This expanded the international representation in the curia, but also pushed the the episcopal office on even more full time Roman officials. The priests who have been named as cardinals since then have been illustrious theologians - Congar, Dulles, Grillmeier, etc.- who were already over 80 and so ineligible to vote for Pope. Francis has named a few bishops (not Arch-) that adds new wrinkle to the next conclave.