In my (physics) manifolds lecture notes they define vector fields using the product rule, rather than explicitly defining them as derivatives.
Are there any other ways therefore of defining vector fields which are not the usual derivatives that satisfy this product rule?
It is slickest and most efficient to define a vector field as a linear operator $\mathcal{X}: C^\infty(M) \to C^\infty(M)$ satisfying the product rule. However, this is not exactly intuitive, and there are alternate constructions.
One of the most "concrete" of these is to define a vector at a point $p \in M$ as an equivalence class of smooth curves. That is, every such curve determines a unique tangent vector at $p$, namely the tangent vector to the curve. Two curves are considered equivalent if they determine the same tangent vector, and we write $\gamma \sim \alpha$ for $\gamma,\alpha: [-1,1] \to M$ such that $\gamma(0) = \alpha(0) = p$ if, in every chart $u: U \to \mathbb{R}^n$, where $U$ is an open subset of $M$, we have $(u \circ \gamma)'(0) = (u \circ \alpha)'(0)$. (In fact, it is sufficient to check this condition in just a single chart.)
Here's a rough sketch of the picture I drew in Krita: