How to interpret tangent vectors on a manifold as derivations?

You can interpret a tangent vector at $p \in M$ as a derivation in your sense. A tangent vector at $p \in M$ is a map $v \colon C^{\infty}(M) \rightarrow \mathbb{R}$ which is $\mathbb{R}$-linear and satisfies

$$ v(fg) = f(p)v(g) + v(f)g(p). $$

If you consider $\mathbb{R}$ as $C^{\infty}(M)$-bimodule by the structure maps

$$ f \bullet x = f(p) \cdot x, \,\,\, x \bullet g = g(p) \cdot x $$

where $\cdot$ is the regular multiplication of real numbers then the equation above becomes

$$ v(fg) = f \bullet v(g) + v(f) \bullet g. $$

In more fancy terms, you have the evaluation homomorphism $\operatorname{ev}_p \colon C^{\infty}(M) \rightarrow \mathbb{R}$ and $\mathbb{R}$ is a bimodule over itself so by pulling back using $\operatorname{ev}_p$ we give $\mathbb{R}$ a bimodule structure over $C^{\infty}(M)$.