[valarauko]

I would not count Ajivika or Charvaka as within the ambit of Hinduism, and neither did the Ancient Commentators. Both were considered heresies. Indeed, Ajivika is better understood as a distinct dharmic religion. In addition, Advaita is not a "non-sentient God", but that's a separate discussion.

For the sake of the parent question, it would make sense to restrict to the Astika schools. Basal to these schools was an acknowledgement of the supremacy of the Vedas (or at least tacit acceptance of their authority).

[woadwarrior01]

Although most of the Nastika (atheist) literature has been lost over the ages, some texts are still available (The Charvaka text Tattvopaplavasimha in particular, I have a copy of it) and there's no reason to discount their philosophical contributions. Their contributions are no less important than the contributions of the Astikas (the Theists). Also, let's not forget that Buddhism which was more successful abroad than it was in India is a Nastika faith. OTOH, you'd naturally argue that Buddhism isn't Hinduism.

* In addition, Advaita is not a "non-sentient God", but that's a separate discussion. *

One of the Advaitin Mahavakyas (Great sentence/quote) from Aitareya Upanishad is: "Prajnanam Brahma", which translates to: "Knowledge/Consciousness is God". One of the fundamental ontological dichotomies in Hinduism is the difference between "Chit" (things that are conscious) and "Achit" (things that are not conscious). Essentially, the distinction between sentient and non-sentient objects. Although consciousness is a trait of "chit" or sentient objects, consciousness itself is not sentient (that'd be a circular definition). We don't even have to go that far if you interpret it as knowledge, because knowledge by definition non-sentient. Extending from that, if God is consciousness and consciousness is non-sentient, then God is non-sentient.

I know that the Adviatins try very hard to paper over this philosophical wrinkle, but that's what drew me to their school of philosophy. A religion with a non-beneficent and non-sentient God is beyond cool, IMO. One of the taunts that Ramanuja (a Vishistadwaita philosopher) had for Advaitins was to call them "Prachanna Bouddhar" (Closet Buddhists). Because from his theistic perspective, the Advaitins with their non-sentient God are only a hop skip and a jump away from the Buddhists (who were originally atheists).

[valarauko]

My concern is not whether the nastika schools had valuable contributions to the advancement of Indian philosophy. The question was one of defining Hinduism. Within the ambit of that question, casting the net wide enough to include Ajivika and Charvaka is too broad, and one the Ancient Indians wouldn't likely have made either. For example, the Hindu acharyas were pretty clear that Buddhism and Jainism were heresies and beyond the pale, and their opinion of Charvakaism is well known. Therefore I suggest that the question is best answered by restricting to the Astika schools, a categorization the ancient commentators would likely agree with.