In love and jealous and they have not made a promise. This typical kdrama plot is my guess.
I don't think he loves her or is jealous, It's more of the same reason why she can't bring herself to tell DJ and also her family members. I think its more of like, How would you explain that you are in love with the child of someone that "ruined" your family to your family and also because he knows she will end up being hurt when everyone finds out