Short answer: The developmental process starts from oogonia as follows:
Oogonia(7million) in the foetus develops into primary oocyte(2million) in the foetus prior to gestation which again develops into primordial follicle(700,000-800,000)at birth.
From the second excerpt of your question I could understand that it said,
"each ovary contains approx. 400,000 follicles at birth." which is definitely correct if you consider that total number of follicles in both the ovaries would give 800,000.
Excerpts from the references that lead to the short answer above:
In the developing female fetus, oogonia become primary oocytes that begin the first division of meiosis. However, this division is not completed and the primary oocytes remain “frozen” in the prophase stage of the first meiotic division.
At birth, oogonia are no longer present. Each primary oocyte is surrounded by a single layer of squamous epithelial cells called follicular cells. The primary oocyte together with its follicular cells is called a primordial follicle. There are about two million primordial follicles with their primary oocytes in the ovaries at birth suspended in the first division of meiosis.
As the female grows, primary oocytes begin to die and disappear with their follicular cells. This process continues until puberty when there are only about 400,000 primordial follicles left in the ovaries. The primary oocytes continue the process of oogenesis after puberty begins.[Source]
The total number of primary oocytes at birth is estimated to vary from 700,000 to2 million. During childhood most oocytes become atretic; only approximately400,000 are present by the beginning of puberty, and fewer than 500 will be ovulated.[Source]
Primary oocytes reach their maximum development at ~20 weeks of gestational age, when approximately seven million primary oocytes have been created; however, at birth, this number has already been reduced to approximately 1-2 million.Recently, however, two publications have challenged the belief that a finite number of oocytes are set around the time of birth.[Source]
In the human embryo, the thousand or so oogonia divide rapidly from the second to the seventh month of gestation to form roughly 7 million germ cells.[Source]