It seems to me thay compersion is maybe one flavor of empathy. Like, you can empathize with someone's anger and get angry for them, you can empathize with someone's sadness and feel sad that they're sad, and you can also empathize with their happiness about love and feel happy for them. The last of which would be compersion. But yeah, it's a lot less specific.
Me, 30ish bi female, been doing solo poly for roughly 5 years. Gia, Clay, and Pike, my partners. Davis, ex/friend/"it's complicated." Eric, Gia's husband. Bee, Gia and Eric's toddler.