I think you are right but I couldn't find it anywhere. I would think the State Funeral would only be for the President or Vice President.
From what I have read, the President can authorize/honor anyone to have a state funeral -- meaning to lay in state in the Rotunda. If I am not mistaken there was a Supreme Court Justice that got one and someone else said that General MacArthur did, as well. Obviously, it goes without saying that the President is always accorded that privilege.
Now, I have a lot of fear with that if a Democrat is president when certain people pass into the great beyond. It could be a very scary thought to some of us.