Atheist is someone who is a person who disbelieves or lacks belief in the existence of God or gods.
A theist is someone who is a person who believes or has belief in the existence of God or gods.
There are many variables of these two positions. You can be a hard (gnostic) atheist ("there is no god and no god exists") or a soft atheist ("I choose to not believe, and this doesn't impact my life" along side its brother, agnostic atheism,"I do not believe, but I do not know if I'm right or wrong"). You can be a gnostic theist ("There is a God, and I know he exists" or a soft theist "I believe but I live my life in a way that it does not impact me much" or agnostic theist "I believe, but I'm not sure if I'm right or wrong".
throw in full blown agnostcism, deism, and you've got a theology salad. You said earlier in the page that there's more ways to be theist, the same is true for atheism. Atheism does not mean that the person claims there is no God.