I should add that Jennings is developing abstract projective geometry on the basis of perspective drawing; motivating concepts such as "point at infinity", and indirectly, the zero value coordinate in ideal points and ideal lines. If there is a formal mathematics involved it is called projectivization. It is (I believe) the mathematics used in Persistence of Vision Ray-tracer http://www.povray.org/ and robotics http://www.ipb.uni-bonn.de/html/teaching/3dcs-ge-2020/stachniss/2020-3dcs-02-homogeneous-coords-4.pdf
My question is about the mathematical formalism of this particular aspect of "concrete" projective geometry. There may be no consistent answer to this; since it is apparently not considered an essential topic in theoretical projective geometry.
Does the mathematical formalism of perspective (central) projection assume all scene points are in or beyond the view plane, or at least "in front" of the central point (i.e., eye)?
The excerpt below is from Modern Geometry with Applications ,by George A. Jennings. His pictures do not appear to agree with his words. For example
Only one radial line in the plane $\overline{OP}$ does not connect a point on $L$ to the eye.
For that to be true, we would have to include points of $L$ "behind" the eye.
Jennings is clearly appealing strongly to intuition, which is exactly what I like about his book. In other places his mathematical statements are typically correct and transcend his heuristics. Am I to assume the same in this case?
No.
Projective geometry typically uses homogeneous coordinate vectors, where you may multiply all coordinates with the same (non-zero) scalar factor without changing the point it represents. Using a negative factor exchanges a representative "in front of the eye" with one "behind the eye".
In a less coordinate-based view, the points of projective geometry correspond to lines through the origin one dimension higher. Lines, not rays, so the part behind is just as valid as the part in front of the central point, the origin.
There are certainly variations of this. Situations where you employ a machinery very much like projective geometry but talking signs and rays into account. Such a system would not be considered standard projective geometry, though.
However you could also take the thought from the opposite direction. Given that every line though the origin represents a point, where in the viewing plane does that represented point lie? Well, just intersect the line with the viewing plane, and you get a represented point that lies in the viewing plane. An intuitive setup would have that viewing plane in front of the eye, so most points would indeed be in front of the eye.
The one notable exception to this is the situation where the line representing a point doesn't intersect the viewing plane because it runs parallel to that. Then you get a point at infinity, which can't be represented as a finite point in the viewing plane.
In your diagram, the line $OL$ plays the role of the viewing plane. Lines or rays through the eye represent points on that line. So take care to not assume a viewing plane vertical in front of the eye in that diagram.
Your edit changed the question from “projective geometry” to “central projection”. This shifts away from a rich and well defined field of mathematics, where many things have specific meaning to make sense in combination, towards a specific operation that has far fewer strings attached and thus far more leeway to be used in whatever way makes most sense in a given situation. So for example a ray tracer indeed traces rays, not lines, and might even discard parts of the scene that are in front but too close to the eye. When it comes to ideal points and infinity, central projection using rays might be useful for building intuition but less suitable for a rigorous framework. Identifying “opposite” ideal points is a critical aspect of projective geometry.