“From page 3 it looks like there are different rules for churches vs. other religious organizations. I didn’t dig deeply enough to figure out what the difference between the two is.”
Very different rules. Churches are (almost)never required to file, even for the 501 C (3) status. That is how Jesse Jasckon & Sharpton get away with what they do.
As a CPA I was one of the leading experts in the country on this subject for years and was on retainer with several major religious groups (denominations) as a tax consultant.
My 501(c)(3) just has to e-file the Form 990-n. Nothin’ to it, since we get less than $10K/yr in contributions.