I think rin was actually a spy from hidden mist village, during the war her mission was to sneak info out of leaf village to the mist village. But her attachment to kakashi and her guilt towards obito made her to betray the mist village. If you see the manga the guys surrounding kakashi and rin are wearing hunter nin masks who's sole purpose is to hunt down betrayers and dispose their body of in order to prevent other villages from stealing intel and techniques (Check the first encounter arc with zabuza when naruto was young, kakashi himself explains the role of hunter nin's). The Hunter Nin's wanted key intel from rin before killing her, her only choice since she was surrounded made kakashi promise to kill her in order to protect the village. That's why the ninja's were so interested in rin.