So, would "karuna" be love with the intention to set things right, while "metta" would be love with the intention that things continue along in their rightful manner?
All Buddhist doctrine has it's roots in the Four Noble Truths, the fundamental exposition of the nature of samsara; its arising; its cessation; and the path that leads to that cessation. That nature is suffering.
The antidote to any suffering is compassion. So that's Karuna in a nutshell.
Compassion is not just an intention though. For enlightened beings, it is the nature of their action, of their intention, and of their view - or we would say, body speech and mind. So not just intention.
Maybe example - Someone loses their house in a fire. You have compassion for them and want to relieve their suffering because you empathise. You have an idea of what that would be like, and you feel compelled to help, as you would like to be helped.
So you've heard Buddhism is a path. Then compassion is the nature of that path, because that's all that Buddhism is - a path to end suffering. If you help others to be free from suffering, then you also have empathy for yourself. 'How can I let myself suffer like this? This is not right, I must do something.' Compassion becomes your path, you stop ignoring yourself and find out what it is you really need to be happy. What you need is the same as what others need. So compassion is the most important.
Here's a translation of a Tibetan stanza on the four immeasurables
May all beings have happiness and the causes of happiness
May all beings be free from suffering and the causes of suffering
May all beings never be separated from the happiness that knows no suffering
May all being abide in equanimity, free from attachment and anger that holds some close and others distant
Metta is the first, giving happiness to others
And karuna is next, taking suffering from others
Then mudita, sharing in the happiness of others (like not being jealous of other's happiness or overly involved in your own suffering)
And upeksa, not being swayed by circumstances of others, such as not being appreciated when you are kind, or being taken advantage of when you help. This is the unattached emphasis, so you don't get pulled back into the swamp of samsara when you try to help others.
The four immeasurables balance each other like this. They're simple, but simple meditations are often the most profound.