Right--I'm not sure why they didn't include "have...on" as one of the translated definitions for "har...på". It's natural in English to say "I have a jacket on", too. Maybe they don't care for the terminal preposition structure.
I think you are looking at it wrong. Although it confused me at first since you are right in saying har ... på is to wear, i believe it uses the har as i have and the pa and on. Still don't know how to tell the difference