Ok. Let’s back up.
Nomenclature overview: (we can argue about whether changing all the nomenclature is a good idea later. For now, lets just try to both get on the same page.)
The New list is topics you’ve never seen before (ie you haven’t seen the OP). If you have the new setting on its (imho, a pretty terrible) default this list clears every session, and will only show new topics since your last session.
Unread posts are ones that have been updated since you read them and you’re “tracking”. By default, you’re tracking posts that you’ve spent 2 minutes in. I’ve changed this to 1 minute for myself, but I’m considering changing it to “always”.
When you watch a post, updates too it show up in the upper right corner, where notifications for replies and likes normally are. If you’re not logged in, they also get sent to your email. These posts also show up on your unread list.
My workflow is to go to the unread list, open all of those in tabs, and read them. Then I go to the new tab and open all the ones that look interesting, and read those. Some topics, like this one, I track because I want to be an active part of the discussion. It works pretty well for me, even coming from a vBulletin background.