It is this "they do not believe in the existence of God", that is what I am saying is an active belief. It is an postive affirmation of not believing in God. They would not self-identify as atheist if they had no opinion. They clearly have an opinion as you just stated, "They do not believe in the existence of God".Thus, the person mentioned could accurately be classified as a "weak atheist" ..... as they do not believe in the existence of God or gods.
"Belief is the state of mind in which a person thinks something to be the case, with or without there being empirical evidence to prove that something is the case with factual certainty. Another way of defining belief is, it is a mental representation of an attitude positively orientated towards the likelihood of something being true.
[From Wiki].
To not believe something, is a belief. It is my belief that little green men don't exist. Do I know that absolutely, no. But saying "I don't believe they exist", and saying "I believe they don't exist" is saying the exact same thing.