Atheists reject that God exists and agnostics do not know if God exists. What is the word for people who do not care if God exists? Does some word moonlight for that or is there a specific term (apart from the usual adjectives like indifferent, etc.)?
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There is a portmanteau word apatheism (combining apathy and theism/atheism) which may suffice, providing the agent noun apatheist.
The Wikipedia article lists this indifference as pragmatic atheism, practical atheism and apathethic agnosticism, any of which may be preferable to a portmanteau neologism.
An additional category that may fit the "I don't care" category—besides the apathetic position that "I am bored by and lack interest in religion"—is the position that "the central subject that religion purports to address (namely, God) is inherently incomprehensible and thus irrelevant to human life." This view has been elaborated under two terms: theological noncognitivism and ignosticism. Wikipedia has entries for both terms, as well as for the umbrella term nontheism, which seems to apply to any theory that denies a positive faith in God's existence.
Fundamentally, the theological noncognitivist and the ignostic deny the importance of theological questions; they "don't care" because they have concluded that nothing knowable or meaningful lies at the core of religion to care about.