The complete text of 47 CFR 64.1200, the federal regulations that existed at the time regulating unsolicited phone calls, is here, but the relevant sections that applied to this case are:
(a) No person may: [...]
(c) The term telephone call in Sec. 64.1200(a)(2) of this section shall not include a call or message by, or on behalf of, a caller: [...]
(d) All artificial or prerecorded telephone messages delivered by an automatic telephone dialing system shall:
The wording is important. Section (a) prohibits parties from making phone calls using a pre-recorded voice. Section (c) says that non-profits are exempt from the blanket ban in part (a). But then section (d) says that "All artificial or prerecorded telephone messages" must include a return phone number -- in other words, even if a party is allowed to make pre-recorded calls at all, they still have to conform to the restrictions in part (d).
I think this clearly applies to non-profits as well, for two reasons:
In her ruling in case Y32174, Judge Garrow said:
"...subsection (c) excludes telephone calls initiated by a tax-exempt non-profit organization, so therefore they are not restricted from initiating telephone calls to a residential line without prior express consent of the caller. And therefore, because they're exempt, it would make no sense to have subsection (d) apply."
but this is exactly wrong. Because non-profits are exempt from the complete ban on pre-recorded calls, this is the only way it would make sense to have subsection (d) apply, requiring them to include a phone number. If non-profits were banned from making pre-recorded calls, then it would make no sense to have subsection (d) apply -- a law can't tell someone, "You may not make any pre-recorded phone calls... but if you do, please include a return phone number." Subsection (d) says that there are some parties that are not banned entirely from making pre-recorded calls, but there are still rules that they have to abide by.