Fine, then with " strong atheist" you have specified a person who both does and doesn't believe that God does not exist.
Please re-read the thread and if you still want to pursue the contradiction I'll say nothing more.
I'll break it down for you:
Atheist = one who lacks belief in the existence of God/gods
Weak Atheist = one who lacks belief in the existence of God/gods, but does not believe actively that God is an impossibility.
Strong Atheist = one who lacks belief in the existence of God/gods, and goes a step further actively believing that God is an impossibility.
Both strong and weak atheists "lack belief in the existence of God/gods", so they are both atheists. Strong atheists, however, hold the belief that God does not exist.