Here's the distinction as I see it:
Agnostic - Doesn't know whether there is a God or not and doesn't believe that we can ever really know. Will never take a position on whether or not there is a God.
Strong Atheist - Believes positively there is no God. If he sees any evidence for God, he will believe it is flawed because he knows there is no God. Most people mistakenly believe that the majority of atheists believe this, but these types of atheists are in the minority, and in fact, not looked that respectfully upon by many other atheists because making a positive statement that there is no God is very similar to making a positive statement that there is a God - both are just absolute beliefs that rely on faith. When people talk derisively about atheists and describe their beliefs, they are usually talking about strong atheism which is frustrating to the majority of soft atheists...
Weak/Soft Atheist (the majority of atheists) - They believe that there is no God because they have yet to see any compelling evidence of a God. They will keep an open mind as to the existence of God, but have yet to see evidence as of yet. If they are made aware of any compelling reason to believe in a God, then they will certainly believe. If God appeared to them tomorrow, they would cease to be atheists. They don't discount the possiblity of God exisiting, but in the same vein, they don't discount the possibility of the Loch Ness Monster or leprechauns existing. I'm not saying this to be flip. We might find out someday that there were a bunch of reclusive Irish midgets that could do things that regular people couldn't, and we might find out for sure some day that there is some great force out there that is intelligent and guiding us. Until then, soft atheists don't believe in God, and in fact, most are not even capable of believing in God as anything more than a vague and unlikely possibility.
I admit, strictly speaking, to lumping weak atheists into the agnostic category, but appreciate the finer distinction of how open their mind might be to the possibility of a supreme being.
Agree completely with your definition of a strong atheist. It is my conjecture and experience that many in this category have experienced deeply traumatic experiences in their life that lead them to conclude that God can't exist, or that if He does, then He's evil for having allowed the trauma to be perpetrated on them in the first place. Their denial of God's existence is more emotional than objective.