i usually do whine a bit about large photos, but you could upload bigger ones in this case.
lateral run out is the horizontal movement of the ring gear or carrier as you spin it. like sideways back lash.
you'd set up your dial indicator perpendicular to the outside face of the carrier flange for the carrier, and the out side of the ring gear for that. it's one of the first measurements you should take after getting your initial carrier shim stack sorted out. it lets you know if your carrier is bad, the ring gear is warped or not seated right, or the housing is bad. excessive run out will also show up as wonky patterns or not being able to get a decent pattern at all 3 spots on the ring gear you should be checking with marking compound.
your last pattern still looks a touch too deep, i'd make small moves towards the ring gear and tighten up that bl till your pattern is a bit more toe shifted. that's known as a "race" set up. you'll be running larger tires that put a higher load on the axle, you want the bl good and tight because it will force it's self apart, opening up the back lash, when you're on it.
look at louie's pictures. you're after a more oval pattern and you've still got a bit of a half moon going on from the looks of things. make sure you are putting some good resistance on that carrier when checking the pattern.