Should you let PVC primer (purple) dry before applying cement?
No - PVC Primer does not need to 'dry' before use - in fact, it needs to be wet to do it's job and not impede the cement - remember, plumbers in a hurry use this daily and aren't sitting around waiting for the primer to dry before grabbing the cement.

I make sure everything fits first, then grab the primer and swab it on, set it down and grab the cement and swab it on.  If you wait - it's likely no more the 10 seconds.  If you've waited a few minutes because you've gotten distracted, made a mistake or something and it's fully dry, just reprime to be sure the plastic is accessible and softened for the cement.  If you don't- it can prevent the cement from bonding as well to the plastic.  

The cure-times listed on the can are for after the cement cures- typically 2 sets of times will be listed - one for handling and one for pressure testing.