I will likely always call the third continuity the Modern Age, although it is now no longer the current age, and no longer being published. It ran from Crisis on Infinite Earths #11 through August 2011. This is what I grew up with, these are the covers I remember the best, this is what I wish was still being published.
I have read every single continuity, and many books outside of continuity, comic book published by DC Comics in this time frame. I also collected a few series during this time frame, although mostly I didn't get my own copies, just read the collection I had access to.
Even though it has been several years now since this continuity ended (I am writing this in August 2015) this is still primarily what I think about when I think of DC Comics.
The proper name for this continuity is the Bronze Age.