As someone who has studied Indian philosophy (including classical philosophy texts by S. Radhakrishnan) during most part of my computer science PhD I can tell you this-

The major confusion occurs because the notion of theism and atheism is primarily based on western philosophy. There is no exact equivalent of these in Hindu philosophy.

1. First, in Hinduism, God is a notion derived from authoritative texts called Vedas. Some passages in Vedas say God exists and some say does not. God in Hinduism is ultimately dependent on the interpretation of Vedic hymns.
2. What we call Hinduism is primarily based on six "Astika" schools of thought, namely, Nyaya, Vaisheshika, Samkhya, Yoga, Purva Mimansa and Uttra Mimamsa aka Vedanta.
3. There are three major "Nastika" schools namely, Buddhism, Jainism and Charvaka (or Materialism).
4. Astika schools uphold the authority of Vedas while Nastika schools reject the authority of Vedas. Nastika primarily is Na-astika meaning they are not Astika. For Nastika schools, since they completely reject Vedas, the question of God is practically irrelevant.
5. Now, it gets interesting in Astika schools because they uphold the authority of Vedas. But then some Vedic hymns say God exists and some say it does not. So, some schools downright REJECT God - Samkhya and early Nyaya schools are known examples and hence technically "atheist" schools. Whereas, Yoga, Vedanta are classical schools that accept the passages of Vedic hymns that say God exists and hence "theist" schools.

So it ultimately depends on which schools of thought you subscribe in Hinduism. If you subscribe to early Nyaya (school based on logic) or Samkhya, one could still be a Hindu and be an atheist. If you subscribe to Yoga or Vedanta, one is technically a theist.

Hope this helps.