I think the "not caring" association is coming from the second definition. If you are uncertain or indecisive about something, it means you don't have a strong opinion, which could be mistaken for apathy. It might also be a cultural idea that if you care about something, you will form a strong opinion of it. If you haven't taken the time to sort out your feelings on a matter, it must be because you don't care. People might settle for ambivalence when they've given up on an issue. In that way, ambivalence and apathy would have a strong correlation. It is not a term used with a lot of passion in the US, I think.1. The coexistence of opposing attitudes or feelings, such as love and hate, toward a person, object, or idea. 2. Uncertainty or indecisiveness as to which course to follow.