Usually when an atheist says "there is no god" what they mean is "there is no reason to take seriously the claim that there is a god, because there is too little evidence". And by "too little" they mean the same amount as there is for the core of the moon being cheese, or something, i.e. essentially zero.

So I don't think they're really making the assertion, or the mistake, that you think they are.

Theists are the ones claiming an explanation for the nature of existence - the burden of proof is on them. “Belief” is not a valid argument when putting forward a theory. I might “believe” we are living in a simulation, but it doesn’t make it any more true than a belief in a magical all knowing space unicorn that rockets through the universe propelled by rainbows while it creates life or any other as of yet non provable nonsense

> An atheist claims "there is no God" or that eg "there is no Christian God".

A gnostic atheist claims that, and yes that is a faith-based position. I think most atheists are agnostic atheists though.

It's the other way around

https://en.wikipedia.org/wiki/Russell%27s_teapot

Nobody can prove existence or non-existence of a god (except a god if a god would exist).

It is usually just a statement about likelihoods. Absence of evidence for existence of god(s) makes it seem very unlikely god(s) exist. Especially after millennia of people looking for evidence.

Not all beliefs are equal. You personally probably wouldn't put the likelihood of the existence of the great spaghetti monster in the sky to the same level as your belief in your personal god.

I have the proof! I saw his meatball eyes in a dream! All hail the FSM!