While I think there are some contributing factors involved which pertain to the particular subject of god(s) (non-strict definitions of god(s), some theists not grasping disbelief at all, anti-theistic agendas etc.), much of the strange thing that is happening seems to me a language issue.
While technically "I don't believe in X" can mean merely lacking a belief, it's generally taken as some kind of active disbelief. Telling someone that you don't believe them suggests that you think they're either mistaken or lying.
So for many (probably most) listeners, "I don't believe in god" is functionally identical to "I believe there is no god" and therefore describing quite a different position to "I'm unconvinced".
I don't think this is specific to god beliefs, we could exchange in ghosts, climate change or whatever and essentially see 3 conclusions (it's true, it's not true, not convinced either way) and the position of not having any conclusion, which I would separate from concluding that a claim is unconvincing. And while this is a scale and people can be at various positions along that scale, ultimately it would come down to what factors into someone's reasoning or behaviour. So, someone unconvinced by climate change might be fine with clean energy regulations (just in case), whereas someone who thinks it's probably untrue may be opposed.
What David Mo' describes here as definition for atheist and agnostic in philosophy, in my experience also represents most people's understandings of the terms. Those who are merely unconvinced often see the term "atheist" as describing a position they don't hold and "agnostic" as a more useful label.