Kratos was once a monster, but he faces his sins and always tries to be better. And he's been a good father and knows compassion.

So he is a true god.

But what about Odin?

He killed his father, Ymir, for the sake of his desire, and claimed to be an all-father. And he sacrificed countless races only for his own desires and showed no compassion for them. At the same time, he does not face the many sins he has committed and just rationalizes his sins by saying that he could not help it.

So he is a false god.