No, the ordinance does not make "criminals out of anyone who holds a newspaper over her head in the rain."
The anti-camping ordinance is appalling in many ways, not least of which that it criminalizes sleeping outdoors with blankets. Staying warm is a life and death issue, so there are lots of sound reasons to be against the ordinance that criminalizes the homeless keeping warm while sleeping. However, it most certainly does not ban covering your head with a newspaper. That does not fit the ordinance's definition which is specifically about sleeping over or under a bedroll, cardboard or newspapers. Accuracy in language and news reporting matter.
Sec. 8-1-22 Camping prohibited; exceptions
(1) For purposes of this section “camping” is defined as:
(a) Cooking over an open flame or fire out-of-doors; or
(b) Bathing in public for purposes of personal hygiene; or
(c) Sleeping out-of-doors under one of the following circumstances:
(i) Adjacent to or inside a tent or sleeping bag, or
(ii) Atop and/or covered by materials such as a bedroll, cardboard, newspapers, or
(iii) Inside some form of temporary shelter
It took me all of a few seconds to follow the link in the OP through to the source and find this.