In Katz, V. (1998), A History of Mathematics, p.117, we read that early Greek geometers defined conic section as formed by planes cutting the generating line (the hypothenuse of the generating triangle of the cone) at a right angle. The section was then elliptic, parabolic, or hyperbolic, depending on whether the angle at the vertex of the cone was acute, right, or obtuse, respectively. Appollonius on the other hand, let the conic sections be the result of cutting a cone by a plane at any angle; if it cut both legs of the axial triangle of the cone, then the section was elliptic; if it cut neither of them, and was thus parallel to one of them, parabolic; and if it cut one side and the other side produced beyond the vertex, hyperbolic.
Thus, while the earlier definition has only one variable (the angle at the vertex), the latter definition has two (the angle at the vertex and the angle of the cutting plane to the base plane). I've seen it stated that the curves generated by the former definition are exactly the same as that of the latter, with the exception of the 'degenenerate cases' of a circle and a point. In other words, the more `restrictive' definition can already produce all ellipses (excluding circles), parabolas, and hyperbolas.
Is this true, and was this known to Appollonius?