You sort of have to think of changing the processing time as something separate than how you expose, though related.
By how you expose, you can alter the density of the final developed negative... but by how you develop the negative, you can also alter the final density.
Most people push by whole stops, so may ask that 500 ASA film be pushed one stop. Now they may therefore rate the film at 1000 ASA to compensate for the one-stop push to end up with normal density (underexpose to decrease density but push process to increase density.) But some people will ask for a one-stop push but only underexpose a little, like rate the stock at 640 ASA or 800 ASA. This way, the final developed negative is still a little denser than normal.