Close shape and close shape with tolerance are 2 variations on the same core function. Close shape only works on a very specific and narrow condition where you have a nearly closed shape that’s disconnected only at one joint. Meaning that all other nodes between segments of the shape are full joined. As long as the single unjoined pair of nodes are within overlap tolerance, the “close shape” function will join the nodes and close the shape.
The only fundamental difference with close shape with tolerance is that you are able to specify a wider tolerance where the unclosed pair of nodes are not perfectly aligned but this necessitates LightBurn making an actual change in the design: either by moving the location of nodes slightly or by adding a line segment that connects the unjoined pair.
Auto-join selected shapes is similar in that it’s meant to join segments but is functionally different. Auto-join selected shapes will join all the overlapping nodes of all independent shapes that are within the selection. So imagine you have a string of 4 lines aligned end-to-end so that they appear as a single line, but where each individual line segment is not semantically joined to its neighbor. By selecting all these individual line segments and then doing an “auto-join selected shapes” it will semantically join all the individual line segments into a single larger line shape. Note, that this is still an open shape.
You can take this farther by taking 4 lines that align to form the outline of a square, and then auto-joining them. Because they were properly aligned, the line segments individually will be joined and will subsequently result in a closed shape.