OP, you mention ‘let those without sin cast the first stone’ but the whole context of those verses answers the question (I think).
After Jesus said that in John 8, he went on to talk to the woman who was just about to be stoned and this is what happened next..
“They were using this question as a trap, in order to have a basis for accusing him. But Jesus bent down and started to write on the ground with his finger. When they kept on questioning him, he straightened up and said to them, “Let any one of you who is without sin be the first to throw a stone at her.” Again he stooped down and wrote on the ground. At this, those who heard began to go away one at a time, the older ones first, until only Jesus was left, with the woman still standing there. Jesus straightened up and asked her, “Woman, where are they? Has no one condemned you?” “No one, sir,” she said. “Then neither do I condemn you,” Jesus declared. “Go now and leave your life of sin.””
John 8:6-11
You’re right that we shouldn’t be a judge of others but to become a Christian means to literally become ‘a follower of Christ’. Jesus himself said to the woman ‘go now and leave your life of sin’.
In other words, if nothing changes in your life and you continue just as you are and do not commit to literally ‘following Christ’, then that’s the biblical test for who is a Christian.
As for Lily - I guess time will tell!