What is the "type" of a contact vector field? 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 $\mathcal{C}$.
 A: Using notation $X\theta$ for Lie derivative of a 1-form $\theta$ along a vector field $X$, just check that multiplying $\theta$ by a function and then taking these derivatives pops out lower order terms, which are linear combinations of previous entries in your sequence: $X(f\theta)=(Xf)\theta + f(X\theta)$. This proves that type depends only on the contact structure, since the contact forms of the contact structure are precisely the nonzero multiples of $\theta$.
Generalize to any subsheaf $I$ of the cotangent sheaf $\Omega^1_M$ by letting $XI$ be the subsheaf of the cotangent sheaf generated by 1-forms $X^k \theta$ for $\theta$ any local section of $I$ and $k$ any nonnegative integer. For example if $I$ is a vector subbundle of $T^*M$, say with $\theta^i$ a basis of local sections, then any other basis of local sections is $g^i_j \theta^j$ for some invertible matrix of smooth functions $g^i_j$. Therefore $X^k (g^i_j \theta^j)=X^{k-1} ( (X g^i_j) \, \theta^j + g^i_j \, X\theta^j)$ is expressed as a linear combination of the $X^l \theta^i_j$ by induction. I don´t really know how to use this notion of type, since I never ran across it before.
