The main reason your stylus shoots toward rapidly inward when you set it down is because the record has a thick bead on the edge. The stylus is literally falling off that bead (toward the label) and its momentum is carrying it too far inward. It sounds counter intuitive, but to avoid missing the first few notes of music, you should set the stylus down well inside of the bead, not as far toward the edge as possible. It appears you are now doing this, but setting it down just a bit too much inward. I would not try to set the stylus down near the edge, and I would also do what others have suggested, which is to cue the stylus much more slowly. Both slow cuing and not setting the stylus down too far toward the edge should cure your problem; if not, you have some other setup problem (not level, wrong setting of downforce, etc.
Aside from using specialized test gear, to me, the best way to set antiskating is to use something like the old Shure test records that have monophonic music being played at progressively higher volume. At some level, the cartridge will start to mistrack (signal becomes sibilant and broken up). Typically, the side of the groove experiencing lower tracking force will break up first or more severely. If it is the right channel that is mistracking the most, you would INCREASE antiskating in order to bring the right side of the stylus into better contact with the groove. If the left is mistracking more, than antiskating has been set too high. Using this approach, I almost always have found that tonearm manufacturer recommendations for antiskating force are too high -- you usually need less than recommended.
