It appears as though all convex sets are conic. But is every conic set convex? I can't seem to think of an edge case where a conic set isn't convex. Moreover, from the definition of a cone and conic combination:
From the definition of a cone:
Let $\theta \in R$ and $x \in R^{n}$, $x \in C \implies \theta x \in C$.
From the definition of a conic combination:
Let $\theta_{1}, \theta_{2}, ..., \theta_{n} \in R$ and $x_{1}, x_{2}, ..., x_{n} \in R^{n}$, then the linear combination
$$\tilde{x} = \theta_{1}x_{1} + \theta_{2}x_{2} + ... + \theta_{n}x_{n}$$ is a cone if $\theta_{1}, \theta_{2}, ..., \theta_{n} \geq 0$.
Don't these definitions encompass convex sets and convex hulls (restricting $\sum_{i = 1}^{n}\theta_{i} = 1$)?