The facts:
1. The newspaper is from 1994 summer.
2. The photo depicts a small girl.
3. The large headline above the photo is "The Ninth Maiden of Miare Appears".
4. The article says "...the Hieda family announced the birth of a new Maiden of Are".
5. The part at the end of article invites everyone to the birthday celebration for Child of Miare at Hakurei Shrine.
(The wording here is 生誕祝賀 "birthday celebration", which technically can mean anything: celebrating a new birth, a living person's birthday, or anniversary of a birth 1000 years ago.)
6. Akyu's note, "news article from the time of my birth".
7. In Memorizable Gensokyo, the manga that accompanies PMiSS, Akyu said "I've only lived as the ninth [Child] for a little less than 10 years". (Mistranslation in English scans.) Memorizable Gensokyo takes place in 2006 autumn, soon before the publishing of Gensokyo Chronicles.
My current opinion:
None of the convoluted theories are needed, and Zil is right. Even my previous theory about "soul entering the child's body" is unnecessary.
Akyu may simply know nothing about herself being the Child of Miare, until a certain day, when her power and memories awaken. Maybe like the Tibetan tulku, it took a series of rituals and tests for the Hieda to recognize her as the Child.
No matter what exactly happened then, Akyu considers a certain moment in 1994 her true birth.
Regarding Akyu's line in Memorizable Gensokyo, "A little less than 10 years" sounds like an error (over 11 years passed between 1994 summer and 2006 autumn), but it could mean Akyu only formally began living as Child of Miare since 1995 or 1996.
And for all we know, Hieda family could be announcing that "now we know we have a Maiden of Miare", not "this newborn child is a Maiden of Miare".
The birthday celebration can really mean anything. The Hieda family may consider the moment Akyu become Maiden of Miare her birth, too. Akyu's awakening might happen when her birthday was near. The Hieda family might formally recognize her as the Child, or publically announce her identity near her birthday, by choice or by tradition.
The Japanese THWiki thinks Akyu's age to be 11-13 in 2006. I think she's older, at very least 14 in 2006.
EDIT:
Defenders of the "soul vessel" theory told me this is no better than "soul vessel". Well, first, I've put all relevant info together here in one post, so that anyone can make up her own mind; second, replace "Akyu awakened" with "the soul of Child of Miare came to Akyu", and it's really not much different from the soul vessel theory. It could happen naturally, or it could be the result of a terrifically grueling ritual, it's all up to you.
EDIT 2:
The main evidence of the "soul vessel" theory is this line in "Monologue", PMiSS, regarding the Child's physical body:
...it takes nearly a hundred years to prepare my body for the next reincarnation...
Which can be read in several different ways.