Consider following two definitions:
Definition 1. Let $M$ be a smooth manifold, let $p\in M$ and assume that $V$ is a real vector space. We will say that a $\mathbb{R}$-linear map $w:C^\infty(M)\to V$ is a tangent vector at $p$ with values in $V$ if for any $f,g\in C^\infty(M)$ $$w(fg)=f(p)w(g)+g(p)w(f).$$
And we can generalize it to the global objects.
Definition 2. Let $M$ be a smooth manifolds and assume that $E$ is smooth vector bundle over $M.$ We will say that a $\mathbb{R}$-linear map $W:C^\infty(M)\to \Gamma(E)$ is a vector field with values in $E$ if for any $f,g\in C^\infty(M)$ $$W(fg)=fW(g)+gW(f).$$
I think this is a very natural generalisation of tangent vectors and vector fields.
Question. Are there any examples of tangent vectors/vector fields with values in vector bundles which arises in the differential geometry?
Obviously except when $V=\mathbb{R}$ or $E$ is a trivial 1-dimensional vector bundle $M\times\mathbb{R}.$ I am interested if such objects appear in riemmanian, or symplectic, or poisson, or any other geometry.
The notion of derivation is quite general, and encompasses also the definitions you have stated. One particularly fruitful example in differential geometry is the exterior derivative $$d:C^\infty(M)\to \Omega^1(M)$$ taking smooth functions to differential one-forms on the manifold. In the terms of your definition 2, the vector bundle $E$ in this case is the cotangent bundle $E=T^*M$ of the manifold.