Dr. Logan Nelson was the person who mislabeled John Kramer's cancer X-Rays. Due to this, the condition of John was not diagnosed and when it did, it was too late. So, Dr. Logan was not innocent at all.
However, John showed him mercy and saved him when Dr. Logan was severely injured.
From Hollywood Reporter
Apparently, Logan was once a hospital orderly who accidentally mislabeled John's medical X-rays, thereby preventing him from getting timely treatment for his fatal brain tumor.
From this source
Nelson (a former Iraq War medic turned doctor) was actually the man responsible for first mislabeling John Kramer's cancer X-Rays, which led to Kramer's condition not being diagnosed until it was too late. Kramer abducted Nelson and made him a player in his very first prototype game; however, when Nelson was severely injured, Kramer decided to show him mercy (more on that later). As a man dealing with PTSD form his time as a POW, Nelson was a prime candidate to fall under sway of Kramer's brainwashing.