"I don't believe in the existence of God" is not the same as "I believe in the non-existence of God". The same way that "I don't know it is" is not the same as "I know it isn't". It depends on where you put the negation, either on the verb, or on the noun. If you negate the verb, you are assuming no position. If you negate the noun, you are taking a side.
Contrary to what the author in that link wrote, there is such a thing as agnosticism, and she is one.