Let $(M,\theta)$ be a $(2n+1)$-dimensional contact manifold, $\mathcal{C}=\ker\theta$ the contact distribution, and $X\in\mathcal{C}$ a vector field belonging to $\mathcal{C}$. In a couple of minor papers I have found the following notion: $$ \mathrm{type}(X):=\textrm{rank}\{ \theta,X(\theta),\ldots,X^{2n-1}(\theta) \}\, , $$ i.e., an integer $\geq 1$ called "type", which is attached to $X$. (EDIT: by $X^k(\theta)$ I mean the $k^\textrm{th}$ Lie derivative of $\theta$ along $X$.) I found it quite interesting but at the same time doubtful.
The authors of the aforementioned papers claim that it is a contact invariant of $X$ (for instance, $X$ is a contact symmetry iff its type is 1) and use it to obtain certain classification results, but I had a hard time following them.
QUESTION 1: is $\textrm{type}(X)$ indeed a contact-invariant?
(EDIT: the "type" is indeed a function of $p\in M$, and it is globally defined only if such a function is constant.)
In the affirmative case, does anybody know a good reference in contact geometry explaining such a concept with all the due details, including who introduced it in the first place?
QUESTION 2: can the notion of "type" be generalised to more general objects, like $(M,\mathcal{C})$, where $M$ is an arbitrary manifold and $\mathcal{C}$ is a distribution on it?
Roughly speaking, $\mathrm{type}(X)$ should measure how far is $X\in \mathcal{C}$ from being a symmetry of $\mathcal{C}$.