If you accept something with no evidence its a belief, if you reject something with no evidence, its also a belief.
Again, Sam, the truth is that what you apply the word 'belief' to is not really important here. At least not to me. I think you're seriously warping the definition of the word 'belief' here (what you're referring to is called 'thought' in daily language), but I honestly couldn't give a shit.
There are categorical differences between the two.
A theist adds something to their ontology that is not proven.
An atheist does not add that one thing to their ontology.
They are fundamentally different actions. Quite like how putting a cookie in a cookie jar is different from looking at the cookie and then setting it down on the table. If you don't believe me, get a cookie jar. Count the cookies in it. Then pick up another cookie and put it in the jar. You will notice the number has changed. Now pick up another cookie, look at it, and then put it on the table. Count the cookies in the cookie jar again. You'll notice the number hasn't changed. The two actions were categorically different. One had a direct impact on the status of the cookie jar and the number of cookies inside of it. One did not.
The same is true cognitively. Though research on 'belief' is spotty (due to 'belief' being a rather fuzzy concept), the area of your brain that deals with propositions is distinct from that of the area that deals with things already held to be true (in some sense; again, 'true' is a fuzzy definition most of the time).
I think you may be harping on this point because you notice some atheists using the word 'belief' almost as an insult, and you'd like to show that it applies equally to both sides. The truth is that we all function on a massive set of beliefs in daily life. We assume that we are not brain's sitting in vats, or being controlled by evil ghosts, or that all of our friends are lying to us at every turn, etc. Belief is an integral part of life and you wouldn't get very far out of bed without it. 'Belief' is not a dirty word.
But if you really can't understand the difference between a strong atheist and a weak atheist then your troubles are deeper than cookie jars.
(1) I am certain there is no god.
(2) I don't believe there is one. That said, the truth is that I have absolutely no access to such knowledge and I could be wrong.
If you think the word 'believe' doesn't make sense in this sentence then we'll change it to "gorgoflog" or something else that makes you happy. But most of the world has operated just fine with the current definition.
I'll put it one last way.
X = {there is a god}
Theist says X is true.
Strong atheist says ~X(not X) is true.
Weak atheist says X is indeterminate.
If you don't get that, brush up on your first-year logic course.