Last night I finished my second playthrough of Season 1.
I found out that you can indeed kill the Stranger yourself after going through with the amputation. Instead of Clementine shooting him, Lee just strangles him to death one-handed.
It may be because I reacted quickly in the fight because I knew what was coming, or because I killed almost everyone I came across in this playthrough and that somehow unlocks it. Its not a glitch because I kept reloading it and it had the same outcome.