I get what he is saying.
He is saying that there is a difference between claiming to not be convinced God exists, and claiming God does not exist. I agree.
If anyone tells me they are not convinced God exists, I take them at their word. I take your word for it if you tell me that.
But if you tell me God does not exist, which is a knowledge claim I associate with the term atheism, then I will ask for a case.
So again, to reiterate, no case has been made here for the claim "God does not exist".
I am persuaded Christ is God incarnate, Saviour and Light of the world. You are not persuaded of this and that is fine.
I maintain that it is possible, and in fact is the case, that some people are not convinced that Jesus Christ is God incarnate Saviour and Light of the world. You have told me you are such a person and I believe you.
When I use the term atheism, I use it to signify the worldview that God does not exist, not some psychological state of a particular thing. Using it the way some do really trivialize the whole issue, for animals and babies and the like would be considered atheists.
Nor do I think any of us is so much concerned with what people call themselves as much as how people live.
From my experience, atheists aren't so concerned with the fact that some call themselves Christians, but that some, who by their actions, hinder them from doing whatever they want to do. If Christians never acted on their beliefs and convictions, I doubt atheists would be concerned about what they called themselves. It's when atheists feel wronged or persecuted, that they say things like, "who are you to try and stop me from living how I want to?" The underlying assumption in the rhetorical question is that there is in fact, no one who can rightfully stop them from doing whatever they want. There is no objective moral law to which they are accountable.
Practically therefore, the distinction you draw is really one without a difference. For you live as if God does not exist, you act as though God does not exist. All the while saying with your mouth, "I am merely not convinced, so don't read anymore into it than that."