I think both sides look at the same thing in two different ways.

Generally liberals and or atheists tend to believe it means (separation of church and state so called) no government involvement or support of religion in any way.

Conservatives believe it means what the Constitution literally says, in that the government can't pass any laws with respect to religion.

It is very clear that religion i.e. the Bible was an every day part of government and our schools from day one.

Contrast that today where the ACLU and others have made every attempt possible to completely eliminate it from any government entity to the point where they will sue even of a city puts a nativity scene on city hall property or the school wants a prayer before a football game... or even the words in God We Trust on our money.

I don't think that's what Jefferson had in mind.... do you?