Determinant zero means you get a single point of intersection with algebraic multiplicity two. The text you link to speaks of "two equal roots".
In general if a line has an intersection with an algebraic curve (which includes conics) in a point with multiplicity greater than zero, then the direction of the line and the algebraic curve will match in that point. That's one way to define tangent. You can see this from a limit process: take two distinct points on the curve and move them towards one another. As they come closer, the line joining them aligns better and better with the direction of the curve at these points. In the limit, the two points will have become one, and the line will match the direction of the curve in that one point perfectly.
Translated into your formulas, a slightly positive discriminant describes the situation with the two distinct points, and the determinant becoming zero is that limit process. So you see that at zero, you really have two points if intersection coincide. This is distinct from one point staying and one point just disappearing, which would not be possible using a smooth limit process. So a mental image of "just a single point" doesn't do the coinciding situations justice.
A non-tangent line will intersect a non-degenerate conic in zero or two distinct points in general. This is true for the ellipse and in most cases also for the parabola and for the hyperbola with its two branches. It is not true for a single branch of a hyperbola, but in general you'd consider both branches as making up the conic. This corresponds to intersecting the plane not with a single cone, but works the pair of cones you get by rotating lines not rays around the vertex.
A special case is asymptotic directions, as you noted. Let's start with the parabola. If the line goes in the direction of the axis of symmetry, then you only have one intersection, and your "quadratic" equation would actually become linear. You would have the coefficient of the quadratic term become zero. In the formula for computing the solutions, that would lead to a division by zero.
If you do the same kind of limit process for a parabola, moving the line from a generic direction towards the direction where it becomes parallel to the axis, then you would observe one of the points moving further and further away. In the limit you'd have one intersection "at infinity", and the division by zero is the algebraic hint that this might be happening. Projective geometry allows for consistent and well-defined handling of such elements "at infinity". This can avoid a lot of case distinctions. In projective geometry, a parabola would always have two intersections with a secant, one of which may be at infinity.
The same is true for the hyperbola if the direction of the line happens to coincide with one of the asymptotic directions. You'd get a vanishing quadratic coefficient and a linear equation. In projective geometry you would get two points at infinity, one for each asymptotic direction. You might imagine four points at infinity, since there are four asymptotic rays for the hyperbola. But in projective geometry, points at infinity in opposite directions are considered the same, so you'd only get two points at infinity, one for each asymptotic line.
As an extra exercise, you can use the number of points at infinity to categorize conic sections. Zero points at infinity is an ellipse (including circles), one is a parabola and two is a hyperbola. But the set of all points at infinity forms a line in projective geometry. So counting the number of points at infinity is the same as counting the number of intersections with that line. And of course, the single intersection of the parabola again has algebraic multiplicity two. So you can say that a parabola is a conic section that has the line at infinity as a tangent. Neat, huh?