Somewhere along the trail of experience with LB I discovered that the green square node (GSN) was the head or start of the line. I too have had occasional challenges related to the GSN as it concerns open shapes and start of cut, and thus trying to move the GSN such that it was an end of the open shape rather than stuck somewhere in between the ends. The GSN is usually the default start of the cut, although some Optimization parameters can influence otherwise, and there is the “Set Shape Start Point” tool also to affect this; although the latter doesn’t work when the GSN is not an end node on an open shape. There in lies a problem, not your problem, but it’s related.
These things are important to me because I use LB for plasma cutting process, in which its very desirable to control the starting points (as well as direction) of cuts. Open shapes are common in plasma cutting process since the kerf is so large that simply just the kerf can suffice as one dimension of the “shape” in a design with small details, so literally cutting out a closed shape is not always necessary nor desired.
Anyway, I had learned some time ago that the GSN on an open shape can be manipulated by deleting it and thus having it move to the adjacent node in the cut direction, or by closing the shape and then welding or union it with another shape. That suggests to me that LB does some shape node cleanup whenever it has to perform more than single node computations on a shape edit. I have also learned it is much better (more predicable) to close a shape by dragging one end point to the other, rather than by adding a new line and snapping the end points, the latter does not always yield a closed shape that will Fill.
So here’s a tip that I just discovered to move the GSN of an open shape to an end node… First close the shape by inserting (keyboard I while hovering over the insert point on the line) a node near an end, then drag the end to the other end snapping the shape closed, then delete the line you just established in the drag to close the shape but do it with keyboard D command while hovering over the line segment. LB will magically force the GSN to both ends of the renewed open shape !
JW - As an aside, you mentioned that you “added a new line” in the circle area, but even though I can’t seem to separate it from the longer open shape, there appear to be two lines overlapping (based on cut behavior) that are causing the double cut in the vertical segment. The method I describe above solves the problem, but another way to solve it is to use the Optimization param “Remove overlapping lines”.