Can you re-write that without the double negatives?
If you do, you get - A theist believes that God exists.
Which is much more legible.
Which means that you had to move the negative from "don't believe" to "no god" part. Which you can't do when it comes to "don't believe in God"
A person believe/notbelieve in god/not-god.
What you're saying is that not-believe in not-god = believe in god. But not-believe in god is not equal to believe in not-god.
It's a double standard on the use of how you move the negation around.
If not-believe in no-god is the same as believe in god, then by the same principle not-believe in god is the same as believe in not-god.
OR
We can consider "not believe in no God" to be the implicit theism and "believe in God" to be the explicit theism. One is slightly different than the other, just the same way as "no belief in God" is supposedly different than "belief in no God".