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 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 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, the break-up of a romantic relationship, in which the participants do end it.
The fact that the thing being broken into is a compound noun (the middle of a piece of music) might throw some people, so it's easier to just consider...
The announcer broke into the music
Some native speakers (me, at least) would be happy with broke in on the music, because to break in is an established phrasal verb meaning to interrupt. It doesn't necessarily need to be followed by a noun phrase identifying what was broken into (often, it's just an implied ongoing conversation). But if the "thing being interrupted" is specified, it should be introduced with its own preposition (i.e. - on).
Personally I see no real syntactic difference between break into the music and break into the house (the "integrity" of the music/house is broken/violated in much the same way). So it's worth noting that Google Books claims 265,000 hits on that link, as compared to just 168 results for break in the house.
Best Answer
Not sure if break down is really being used phrasally here ...
You can add a preposition after a verb to indicate the direction that the activity was pointed toward. This can overlap with phrasal verbs.
And sometimes the direction of the activity is specified but the subject of the activity is not directly "aiming" the action:
But break down can be used to describe destruction for things other than walls:
It also can be used to not necessarily mean "to be destroyed" but something like "remove". So barriers in the sense of things that block communication are a logical type of wall and to "break them down" means to find a way through the barrier.
It also can mean to take apart in preparation to be put away:
A meaning of break down that is not related to the above, where it 100% is a phrasal verb, means to reduce to smaller parts, typically to try to understand.