Am I a sinner because of the sins I actually commit?
No. You are not a sinner because of anything you have done in actively performing a sinful act. You are a sinner because you are a child of Adam. Being a child of Adam, your old Adam nature says the opposite of what God says.
Les teaches that a person is not a sinner today because of anything they have done so far as actively performing a sinful act. Rather, a person is a sinner because they are a child of Adam. Being a child of Adam, everything God says to do, old Adam says to do the opposite; everything God says not to do, old Adam says to do it. Les notes that millions around us are good people—beautiful homes, beautiful families, well-behaved children, never cheating anyone, never unfaithful—and from all outward appearances they are good people. Yet the sin nature is inherited from Adam, not merely the sum of committed sins. Les was trying to think of a good, simple, yet profound illustration of this old sin nature.