Tonight, at a very good Bible study time at church, a discussion was raised regarding broken hearts. My own impression at the time was that some of the differences expressed were due to using the phrase 'a broken heart' from different perspectives. It wasn't until I got home that I realized that that one little phrase 'a broken heart' could have (at least) 3 different meanings, depending on context:
1. A heart broken by the hurts, pains and harmful things that are endemic in this sin-wracked world. This is essentially a damaged heart. This broken heart is the effect of sin by one person (or persons) against another. This heart needs healing in Jesus, and you wouldn't ask to have your heart re-damaged,
2. A heart broken in repentance. This is the heart that looks at it's own sin, and is broken by how it has disappointed and grieved God. An example of this usage is in Ps 51:17: "The sacrifices of God are a broken spirit; A broken and a contrite heart, O God, You will not despise". This heart needs forgiveness and restoration. If you are intent on growing in Christ, you might very well ask for the discernment of your own sin that would bring rise to it, so that your own hindrences to growth can be dealt with.
3. A heart broken by compassion. This is the heart that sees the sin in the world, and its effects, and is moved to bring healing, restoration, and the Gospel to those under sin's thrall. This is what I believe is referred to by the line "Break my heart for what breaks yours" in the Hillsong United song "Hosanna". This is not a request to damage my own heart, but to bring me to the same compassion that God has for the world.
This is rather the nature of human language (or at least, the way human language is used - I really don't want to get into the Prescriptivism/Descriptivism wrangle here). Words don't always carry a precise, technical meaning. This is why, for example, you may find multiple definitions for a single word in a dictionary - different meanings apply in different contexts (and I'll parenthetically note that it can be dangerous to look at list of meanings and choose one that matches what you want or would like for a sentence to say. You have to examine a sentence to discern which meaning was originally intended). Phrases, even more so.