The ads are then basically a 'template' and as your browser renders the ad, the code reaches into cookies on your computer that have your name in them, and inserts what it finds into the 'name' field in the template.

This is done dynamically (at time of page load), IOW the ad didn't originally come through from the internet/DU with your name in it. Your browser built that ad, in a sense, locally on your PC (or phone or whatever).