There are two related but different notions of "light cone" (or "null cone"):
one in the spacetime [which Minkowski introduced in 1907/1908 as part of his "Space and Time", which introduced the "spacetime viewpoint" and the various terms we use today in relativity], and the other in the tangent space of a spacetime event.
Light Cone in the Tangent Space of an event in Spacetime
The Minkowski Metric on vectors in the tangent space divides the vectors into three types via the sign of the square-norm of the vector. In the (+---)-signature convention, we have timelike, null [or lightlike], and spacelike.... corresponding to positive, zero, or negative. It is said that the light cones encode the causal structure of the spacetime.
The light cone in the tangent space would be defined by the set of all of null vectors (the "lightlike directions"), which would be tangent vectors to geodesics in the spacetime traced out by light signals to or from an event. The tips of these vectors lie on a conical-shaped hypersurface. This cone can also be thought of as the asymptotic surface of the hyperboloids of (say) the future unit-timelike vectors [such a vector is called the 4-velocity, which is tangent to a timelike curve in spacetime that represents the worldline of a particle with nonzero mass]. The light cone therefore represents a finite limiting but unattainable speed for such particle worldlines.]
Some might consider the interior of the cone, which is determined by all of the timelike vectors. Sachs and Wu (in their "General Relativity and Cosmology" AMS 1977
and likely in their "General Relativity for Mathematicians") refer to this as a "solid cone".)
Light Cone in spacetime
The light cone of event O is the set of events reached by lightlike geodesics from O into the past and into the future, which describes the spacetime evolution of a flash of light. (This was Minkowski's use in his "Space and Time" presentation in 1907/1908.)
In some cases, it will look like the typical light cone as we move away from O. However, in some situations the light cone can develop caustics.
For more information, consult, e.g., "Gravitational Lensing from a Spacetime Perspective" by Volker Perlick https://arxiv.org/abs/1010.3416 and https://link.springer.com/content/pdf/10.12942%2Flrr-2004-9.pdf
The interior of the future light cone can essentially be thought of as the "chronological future" reachable by timelike geodesics from O,
although there may be subtleties in tying down a more precise definition.
(See "Techniques in Differential Topology in Relativity" (1972) by Roger Penrose