Sorry I blinked for a couple days there.
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I am awaiting I can share a diagram called The Good Church. Meanwhile see Mandatory Exception at Title 26 USC 508 (as opposed to 501(C)(3)). I have just liened all the assets of The Good Church. The diagram though, it helps clarify this distinction between public and private by a gob or two.
(c) Exceptions
(1) Mandatory exceptions
Subsections (a) and (b) shall not apply to?
(A) churches, their integrated auxiliaries, and conventions or associations of churches, or
(B) any organization which is not a private foundation (as defined in section 509 (a)) and the gross receipts of which in each taxable year are normally not more than $5,000.
There are conditions of private and public where POMC's function reliably, for example.