If Katherine loved Damon it would be in a playful, lustful way. They are too much alike and I'm beginning to see the differences and similarities between them. They both take advantage of people and can be quite pushy as they manipulate the situation around them. Damon learned that from her as they spent time hunting together. They have the lust drive within them because when they want something they go for it without thinking of the consequences. Damon was kind of that way before he knew what she was.
In his past life, Damon pursued Katherine and she rejected him. She was with Stefan at the time. He captivated her presence and aroused a part of her soul that touched her heart. She just wants him back in her life to feel that magic. She's literally living in the past and wants to recapture that love. Like any other women she wants to be loved and know she's coming home at the end of the day to the man she loves. But what makes her different from other women is that she's obessed about Stefan and the love they share. She won't let it go till she gets it back. If she truly loved him, she would let him go. However, she is not that kind of women. She wants him not Damon.
Damon was just her playful buddy who offered her excitement and thrill. Damon is no Stefan. He does not speak in sweet words that soothe her heart. Katherine instead loved him as an exotic lover; not as someone who she wanted to spend the rest of her life with.