John Kramer, the guy in the black and red robes, is Jigsaw. When John's ex-wife, Jill Tuck, was pregnant with their son, Gideon, John built a cute little doll for him. After the miscarriage of Gideon, his divorce with Jill, and his failed suicide attempt, John dedicated his life to help people value their lives. He then updated the doll into the puppet we see now. James Wan, the creator of both the puppet and the Saw franchise, named the puppet Billy. So therefore the puppet's name is Billy, not Jigsaw.
That's not the point of the games. And if those people died, it is their own faults for doing wrong do get there in the first place, and there own choices made during the tests caused them to fail.