That would be an extremely reductive definition that doesn't really tell us much about how caring for others is actually experienced and how it manifests in the world.

Exactly, that's my point.

How would this for example explain sacrificing yourself to save another person, if the very core of caring is to create positive emotions in yourself?

In this case it would be about reducing negative emotions, choosing the lesser of two evils. Losing a loved one and/or having to live with the knowledge that you could have saved them but chose not to can inflict massive emotional pain, potentially for the rest of your life. Dying yourself instead might seem outright attractive in comparison.

this idea that caring is in its essence transactional

That's not actually how I'm seeing it, and I also don't think it's a super profound insight or something. It's just a super technical way of viewing the topic of motivation, and while it's an interesting thought experiment, it's mostly useless.