Hate to break it to you guys, but the person who gets the killing blow gets the toolkit to disarm the mine in the next field. However, I'm still not sure how the title is handed out. The person running the controls up top got it in our run, and I got it myself. However, the tank who also got a few killing blows and disarmed some, did not get the title. Also the person who disabled the field just before the boss fight didn't either.
This is not true.
It is the closest proximity to the Droid, as others have stated.
If you are standing directly under the droid (assuming others are further), you will get the toolkit and, if you then complete the boss fight, the title.