In episode 2x20... Damon tries to get the whereabouts of the vampire and werewolf up for sacrifice from Katherine (when she was under house arrest in Alaric's loft)...
At first she only gave out that she was not the vampire Klaus was going to sacrifice but it was Caroline. As long as it didn't concern her safety she wasn't willing to give away anything...
All it took to make her tell was...
"What if told you that Elena has vampire blood in her system... Imagine how much fun that would be competing with Elena for Stefan's love for, I don't know.. FOREVER" 😉
If I had to come up with a reason why Katherine got Stefan to save Damon it's because...
Plus everything Stefan did to save his brother would go to waste...
Mine is just a simple theory... All though with out doubt Katherine claimed that she loved Damon in 3x09. But then again who doesn't love their pets... considering Katherine treated him like one in 1864.
It's always been hard for me to believe that Katherine "truly" loves Damon too... She has a past with him. May care enough to help out and save him a couple of times... but I won't classify that as love on her part.
Don't get me wrong I absolutely love watching Katherine and Kathmon... but I would hate for them to go down that road. Katherine always looks out for herself first... Being a huge Damon fan I want Damon to be loved the same way he loves.😃