Yeah, that’s my take. “I’m sorry” is about how you feel. It’s like the OP post: you’re expressing sympathy, or remorse - an emotion. If it’s because of something you did, it does double duty as an apology. “I apologize” gives no indication that you’re sorry about what you did, or concede that it was wrong. It’s merely a recognition that harm was done, and giving a concession to the other person to achieve harmony.