I have never seen LP live, but I saw a lot of live recordings of their shows. It´s hard to choose some best LP live moment because their shows are amazing and everytime they surprise crowd with something new, but for me the best live moment is when I see how chester put all his energy and emotions to song (for example Brno 2008 - Given up - how chester just fall on knees and scream on bridge, Smoke Out festival 2003 - Breaking the Habit -you can see how hard it is for chester to sing that song,......) I really like these moments when I see how much energy and emotions Chester put in songs.
And second thing that I like on LP shows is, when you think that the song is ending and then they surprise whole crowd with incredible new outro (for example Webster Hall 2007 - Faint outro)