It's not a lie. That's how the very inventor of the term "separation of church and state" defined it and how it has been used for hundreds of years after. Roger Williams, who first used the term, was a puritan, later baptist, pastor. What you're describing would be an innovation and expansion of what separation of church and state means. It seems what you really want is a separation between religion and state. That is a totally different issue. In the future it would be advantageous for you to be clear on that and explicitly say that it's religion, not just church, that you want to exclude from the state.

I'm curious though. Which people would you want to exclude from the government. Church goers are obvious. But what about the "spiritual but not religious" people? Or people who believe in ghosts but not god? What about natives who believe their native religions? The non god believing buddhist? Who would define if someone is religious enough to be disallowed from running for office? Whoever that person was, that person would get immense power.

Am I understanding you correct that you think no person with religious beliefs should be allowed to hold office?

E pluribus unum

As long as none of the pluribus hold any religious belief?