I'm reading the book "Projective differential geometry: Old and new" and encounter this problem.
Given a projective structure on $S^1$ (or $\mathbb{R}$), we have a developing map $\phi: \mathbb{R} \to \mathbb{RP}^1$ (as in that of a geometric (G,X)-structure) and vice versa. The book claims there is an action of $Diff(S^1)$ on the space of projective structures on $S^1$. I suppose the action of $f \in Diff(S^1)$ is the precomposition $\phi \circ f^{-1}$ of the developing map. This should be fine.
The book also says whenever we have a differentiable action of $Diff(S^1)$, we should have an induced action of $Vect(S^1)$, the space of vector fields on $S^1$, which is the Lie algebra of $Diff(S^1)$. What exactly is the action of $Vect(S^1)$?
If a Lie group $G$ acts on a manifold $M$ then the Lie algebra $\mathfrak{g}$ does not act on $M$, instead we have a Lie algebra homomorphism $\mathfrak{g}$ to $\mathrm{Vect}(M)$ $\xi\mapsto\tilde\xi$ where $$ \tilde\xi_x=\left.\frac{d}{dt}\right|_{t=0} \exp(-t\xi)\cdot x. $$ So in yr example, $X\cdot\phi$ should be a tangent vector at $\phi$ to the space of projective structures.
This is much easier to think about if we follow the book and identify projective structures with Sturm-Liouville operators which is an affine space modelled on the vector space of Sturm-Liouville potentials (thus quadratic differentials on $S^1$). The tangent space at each projective structure is therefore canonically identified with the space of quadratic differentials. Now $\mathrm{Diff}(S^1)$ acts on the Sturm-Liouville operators by change of variable (which amounts to formula (1.12) in the book) and differentiating this gives formula (1.13) which should be viewed as the value of the vector field $\tilde X$ at the point $d^2/dt^2+u$.