In which case we are working of significantly different definitions of justice/vigilante systems.

There is no definition of "vigilante system" that holds such systems accountable to society. If society were to hold such a system accountable it becomes codified into a true judicial system and is no longer "vigilante". These are inescapable definitions that are inherent to the very things they describe.

Then it sounds as if they are one in the same, like two sides of a coin being flipped throughout history.

Legal systems become corrupt and unaccountable, giving rise to vigilante systems. Those that survive long enough begin to improve as those involved seek to better the new systems. Eventually they reach the point of being the new legal system which slowly corrupts itself til it is beyond being held accountable, at which point it reverts back to a vigilante system that competes with new ones.

Perhaps the only issue is we should consider a different name from a vigilante systems that arise due to a corrupted legal system not meeting the needs of the populace vs the legal system itself that falls into a state where it is no longer held accountable.