Focus-directrix definitions of degenerate conics In this answer, the focus-directrix definitions of the degenerate conics are provided, all except for the case of parallel lines, which is the only remaining case.
How is that case obtained in the contexts of both the cone-plane intersection and focus-directrix definitions?
Also, considering a conic section as just an algebraic curve, are there any other conics apart from the usual three and the four degenerates? Apparently a plane can also be considered as one; I'm looking for similar cases that may seem trivial.
 A: This answer is strongly based on my background in projective geometry.
In a projective setting, conic sections can be categorized by the number of intersections with the line at infinity. A pair of parallel lines intersects the line at infinity in a single point. So by that argument, the pair of parallel lines should be seen as a degenerate parabola. The problem here is that if you envision a sequence of parabolas that converges on a pair of parallel lines in the limit, you have to have the vertex of the parabola move out towards infinity. The same is true if you start from a hyperbola, or from a degenerate hyperbola i.e. a pair of intersecting lines. In each of these cases, moving towards the parallel situation will push the focus, center of symmetry, point of intersection or whatever you care to consider towards infinity.
So in some sense, you may say that for a pair of parallel lines, the directrix is the line at infinity, the focus is one point on that line, and the eccentricity is probably $\varepsilon=1$. But the distance from any point to the line at infinity, or a point on that line, is infinite. And comparing infinities is very tricky business at best. Note that you'd get the same line and point at infinity for any pair of parallels with the same direction. So it doesn't make sense to claim that the focus, directrix and eccentricity define a single pair of lines. They do not contain enough information to do so.
As Blue pointed out in a comment, for the cone-plane intersection, considering a cylinder as a suitable limit case for a series of cones would be the correct approach. While you might define the shape of a cone in terms of its opening angle, for a cylinder that angle would be zero so again you end up in a situation where some of the classical definitions no longer apply in the limit because they do not offer enough information in that case. Nonetheless, you can come up with limit processes that go from cone to cylinder in an intuitive way.
In his Perspectives on Projective Geometry, J. Richter-Gebert categorizes conics up to projective transformations. Note that this does make the classical cases of circle, ellipse, parabola and hyperbola all equivalent. He starts with a generic conic section of the form $ax^2+by^2+cz^2+dxy+exz+fyz=0$ where $[x:y:z]$ would be the homogenous coordinates of the points on that conic. He argues that it's possible to use a projective transformation to transform any such conic to the form $ax^2+by^2+cz^2=0$. Furthermore, the absolute values of $a,b,c$ can be changed so you can get $a,b,c\in\{-1,0,+1\}$. Also the order of the three coordinates does not really matter, and neither does an overall sign change. This leaves him with 5 cases to consider:

*

*$x^2+y^2+z^2=0$ has no real solutions, since $[0:0:0]$ is not a valid homogenous coordinate vector of a point. You can call this a complex non-degenerate conic, since it does not factor into a pair of lines, and with complex numbers as coordinates you can get solutions.

*$x^2+y^2-z^2=0$ is your run of the mill real non-degenerate conic.

*$x^2+y^2=0$ is a single real point, but he also views this as a degenerate conic consisting of a pair of complex conjugate lines.

*$x^2-y^2=(x+y)(x-y)=0$ is a pair of real lines.

*$x^2=0$ is a single line (of algebraic multiplicity two).

He explicitly excluded the $a=b=c=0$ case, but if you don't do that, then you'd get to consider the whole plane to be a conic as another alternative.
He also points out that sometimes it is worthwhile to keep an eye on the dual conic, i.e. the conic not as a set of incident points but as tangent lines. For the non-degenerate conics (both real and complex) one can be derived from the other. In the case of a pair of distinct lines, the dual conic implies that all tangents must pass through the point of intersection. But for the double line, the dual representation could allow for either a single point, or a pair of real points, or a pair of complex conjugate points, through which the tangents pass. In this case the primal conic is not enough to derive its dual. I'm mentioning this to complete the listing above, but also because it's another of those situations where your typical definition of an object suddenly fails to provide sufficient information once you reach a sufficiently degenerate situation using some form of limit.
To take a more Euclidean look at this, you'd want to distinguish between ellipses, parabolas and hyperbolas. As I said in the beginning, you'd do that by counting (real) intersections with the line at infinity. The complex non-degenerate conic never has any real intersections with any line. The real non-degenerate may have zero, one or two intersections. The single point may be incident with the line at infinity or not. The pair of lines may have one line be the line at infinity, or may have the intersection at infinity, or have a finite point of intersection. And finally the single line may or may not be the line at infinity. Going by this classification, you'd have $1+3+2+3+2=11$ kinds of conics, not counting the whole plane case nor the distinctions you'd get from also considering the dual conics.
As you can see, it all very much depends on how you count things. What do you consider a conic, and which conics do you consider to be the same type? Above are some of my favorite answers to this, but others with different backgrounds have some very different answers for often very valid reasons.
