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.
When you lead X {infinitive}, you are saying that you are causing X to do or be in a state of doing the verb identified in the infinitive.
When you lead X into Y, you are saying that you are causing X to be moved (i.e. bringing X) inside of/within/surrounded by Y - that's what the preposition in/into means.'
With lead X into Y, Y can be a gerund, which makes it mean the same as lead X {infinitive}, but possibly with a slight implication that X did not originally plan to do Y. This distinction, if it exists, probably matters more in written text, like in a story, than in conversation.
We followed the map which led us to walk down this path.
That crazy squirrel led us into walking down this path.
Best Answer
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...
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.