I agree that it appears that some lists depend on user activity in forum, while others don't.
When you look at New Posts, is it a very long list of topics? Maybe it has nothing to do with user activity. Maybe it only shows posts made today? I thought since you said the New Posts looks like Active Posts, that you must be seeing a 3 page list, like I see in Active topics. That's how I came to my theory. But my theory doesn't explain the 40 page list of Unread Posts. Maybe there is a clue in the wording. 3 of the lists are about Posts, while the last one is about Topics.
Let me see if I can find a guide for the forum, or faq, or something.
Edit
Ok, there is an FAQ link, near top, right corner of page, beside Members and Logout. But I dont' see any info about those lists of posts and topics. Maybe we can find something at the website for phpBB forum software? I'll look around for something...
Edit 2
Ok, I found the Support section for this product:
http://www.phpbb.com/support/ I don't have time to search it right now, but I can look later. I'm thinking their forum might have some info about it.