There is an important difference not explicitly mentioned in those definitions: the grammatical subject of break off refers to a different agent than the grammatical subject of break up.
The subject of break off was previously participating in or causing the activity that ended. The activity ended because the subject stopped continuing it.
The subject of break up was outside the activity, not part of it.* The subject actively interfered with its continuation, causing it to stop.
For example:
The police broke upbroke up the fight.
means that some people, not the police, were fighting, and the police forcibly made them stop fighting. Most likely, the police physically grabbed the combatants, pulled them apart, and restrained them.
The police broke offbroke off the fight.
means that the police were themselves involved in the fight, and stopped fighting by their own choice. Perhaps the police were defending a building against a crowd during a public protest, and the police lost heart and decided to walk away and let the crowd do as it pleased. Or perhaps the "fight" was a negotiation to get a raise in pay. Then if the police "broke off" this fight, that would mean that they gave up trying to get the raise, agreeing to continue to work at their present salary.
When people "break off negotiations", this means that they refuse to continue negotiating. They have given up hope of making a deal, or they demand a concession from the other side before they will even continue talking. They have "walked away from the negotiation table."
*Of course, there is also a different sense of _break up,_ such as [sense 2](http://www.learnersdictionary.com/definition/break), the break-up of a romantic relationship, in which the participants do end it.