Projected points and screen coordinates
It's not 100% clear what the author means here, but I'll choose to interpret "screen coordinates" as "pixel coordinates". These would be related to the projected points by a 2D coordinate transformation.

You're correct that projection is done by dividing each component of a point by its $z$ component. (That's not quite true—actually, we usually use homogeneous coordinates, which have four components $xyzw$, and divide by the $w$ component, which has previously been set to eye-space $z$ by the projection matrix. This frees up the $z$ component of the vectors to be used for the z-buffer.) The projected points $q_i$ are then in "normalized device coordinates" or NDC space, which has a range of [−1, 1] on each axis.

To calculate the resulting pixel coordinates on the screen, we then need to perform a 2D scale-and-translate operation, to map the [−1, 1] range to the desired range of pixels. For example, if we want a 640×480 viewport, we could calculate
$$x_i = 640 \cdot \frac{q^x_i + 1}{2} \\ y_i = 480 \cdot \frac{q^y_i + 1}{2}$$
You might also need to flip the $y$ axis in this transformation, for instance if the NDC space is Y-up but the pixel coordinates are Y-down.