I actually find the use of the word "break" to be very clear, but not in the way that you are breaking ties (or "cutting" ties or "cutting" someone out of your life)
It's VERY similar to agreeing to break a promise. Spoken or unspoken, knowingly or unknowingly, most romantic relationships come with some understandings (or assumptions) and commitments that are treated like promises. Breaking up, to me, implies that you are no longer holding them or yourselves to the promises that defined that model of that relationship (breaking the promise). It then becomes time to make new agreements and expectations. Also notice how the phrase is often preceded by "we", as opposed to "dumped" which is usually presented as a single person's choice to reject another.