Well, these are two completely different commans.
1) Divide - breaks the entity in specified mouse position. The user should define on point.
2) Break/Divide - works based on points, where the entity intersects with another entities. The user should defined segment of the entity.
As it names states, the command allows to the user
1) identify SEGMENT of the entity between two intersection points using mouse
2) either
a) Remove segment(s) of the entity, that is either WITHIN or OUTSIDE of these two intersection points, so the original entities is broken on several segments.
b) Break the entity by segment endpoints (intersection points) - i.e - like Divide command applied for eash intersection point
And here is the video demo:
https://youtu.be/_UHawUJXZe0P.S as for disappering entity after Break/Divide - that's intentional behavior requested by the one of the users, if "Remove Segments" options is checked - the entire entit is considered as one segment and so it's removed.