It was Obitos eye though.
Like it would be weird..
If Kakashi got the eye and it activated because its owner died?
Weird, probably. =D But nothing indicates that can't happen.
Once the Sharingan was implanted, it was Kakashi's from that point on. It was Kakashi's emotions that could trigger MS activation, not Obito's. And these emotions could be connected to anyone Kakashi cared about, Obito's on the list.
Wrong, Specifically said in the manga MS is awakened by one killing the person closest to him
itachi: Shisui
Sasuke: Itachi
kakashi/Obito: Rin
That was one of many of Itachi's lies. It is about emotion.
Otherwise Sasuke wouldn't activate MS - Itachi became a close person to him only after Sasuke killed him, so he didn't technically kill anyone closest to him at that point.
It's emotions - when Sasuke realized who Itachi was, he then got emotions related to that loss and that was what triggered the change.
Not to mention Shisui seemed to be the good guy, and how often do accidents like Kakashi killing Rin happen? Not very often. Chances of Shisui (the good guy) killing his best friend to obtain MS are slim.
One last thing - I agree that actually KILLING the closest person will make the emotions stronger (guilt, etc), so "doing the job yourself" grants better chances of awakening MS than just experiencing someone's death, without influencing it yourself.
But my point still stands - it's about emotion and nothing else.
Edit: By the way, you added Obito to the list, but Obito never killed Rin nor Kakashi. So your argument is proved invalid - it's Obito's emotions related to Rin's death that made him awaken MS, not the action of killing her.