OK! I think I've got it now. It helped to enlarge your video to full-screen size. Then I could see that in the first clip, she seems to jump off her left foot. In the second, she seems to jump off the right. So I thought that it's JUMP you're delaying; of course, then you must delay Walk a little, since if you press Walk before Jump, she goes back to walking.
Anyway, I think experiments show that delayed jump is the key. Here's a video with 3 different delays: left foot, right foot and left foot again. Each jump goes farther because she jumps later.http://www.youtube.com/watch?v=Wdcd7_n1TyU
When I tried these in Lara's Home, each step adds about 5 inches.
This is your discovery; I haven't seen this, because I usually hold Walk, hold Forward and Jump, release Walk and add it back just after she jumps. This gives the same results as the shortest version of our clips. This could give easier set-ups for a dive bug, as I'm sure you intended.