I my opinion ( I do not know about code ) : In wireframe mode we see that when boolean is on - there are created new lines on this top face in main object. At one time new lines connect to the cutting edge and at the other time to the vertex cutting corner depends on position of bool object. Bevel realy is just switch with boolean lines that are created by changing position of bool object. The same thing happens without bevel modifier.

I think lines that are created in main object faces need to be consistant regardless of bool object position. On video we see clear that they connect to vertex or center of edge.

The lines are created in the top face are necessary because Blender doesn't allow holes to exist in faces without those holes being connected to the enclosing face by at least two edges. It is not easy to make those "support edges" get added at consistent vertices as you move things around, unfortunately, and the current code does't even try.

I agree that it is strange that Boolean sometimes creates new vertices in the middle of edges and connects there. I think it is because Boolean works on triangulated faces and then untriangulates them later, not always completely invisibly to the user. I am not inclined to work on making this better in the current Boolean code since I am working on completely new Boolean code that operates differently in many ways -- including not triangulating faces, and using a different algorithm to connect the support edges.

The bevel glitches in the original bug file are caused mainly be clamp_overlap -- because of where the support edges attach. If you turn that off, it will look less glitchy (except for the jumping of the support edge attach positions) but they some edge/faces will overlap I think.

This bug should probably closed as "known issue", which will hopefully be fixed or at least made better with the new boolean code.

great to know that new boolean will be different. Im waiting for it for so long time and propably many people as well.
Can I ask will be finished with new relese 2.82?

If old boolean triangulate faces why vertex on center of edge? when Face is triangulated is cuted in half but from vert to vert.... not from center of line to vert. I do not uderstand this. If is angon is triangulated in different way... but in this case is simple one line through plane so this line will not be atatch to center of line. For me realy werid. :)

No, sorry, the new Boolean will not be in 2.82. Probably not even in 2.83. I thought I was close, but the pesky problems related to numerical issues have proven difficult to get right. I've spent the last 3 weeks redoing an underlying triangulation / intersection library that is core to the new code so that it will be extremely robust in the face of nearly-degenerate geometry. Sorry.

thats ok . Im not developer but I can help in testing and some... thinking out the box. cross fingers for your work. It will be great.:)