Chelia

Maybe Saya was a demented evil individual and Jin saw that which is part of why he hated Ragna for spending so much time with her. Or when she got sick, it made her nuts/evil and when Terumi saw that, he wanted her. As for how she gave him the sword, she could've given it to someone to give to him or placed it somewhere he'd find it. I actually thought of the latter when I found out Saya gave Jin Yukianesa.