Are conic sections obtained from a cone or a double cone? According to Wikipedia,

In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.

However most of the images actually show a double cone instead of a cone and it makes sense to me since a hyperbola has two components.
So is it true that despite its name and definition, conic section is actually an intersection of a plane with a double cone?

Cone/Plane Intersection scenario
A full cone consists of two nappes/sheets. 
Only when the inclinations of cone generator semi-vertical angle $ \alpha$  is more  than sectioning plane angle $\beta$ to the cone axis we get the hyperbola as a real disjuncted double curve.
A parabola or ellipse is produced by intersection when 
 $ \alpha  < \beta$ on a single nappe. 
In this case we are not concerned with generality when  only one cone is partaking and resulting in an intersection.
