A common definition of compassion is “being moved by the suffering of oneself and others, and being motivated to reduce that suffering”. Compassion involves sensitivity to suffering, as well as a motivation to reduce it.
A beautiful, colloquial way I’ve read Christopher Germer describe compassion is this
When love meets suffering and stays loving, that’s compassion.
I find this description so helpful when thinking about self-compassion: when you can meet yourself in your suffering and stay loving toward you, toward all the different parts of you, and stay caring, stay connected to a motivation of care, stay in friendship with yourself…. that’s compassion.