Suppose we have a non-integrable distribution and a riemannian metric on a manifold. Then, one can define the metric using your construction, the resulting object is called sub-riemannian metric. It does not come from any riemannian or finslerian metric. Sub-riemannian metrics are important in the theory of nilpotent groups (and vice versa), as always with such geometric subjects there is a book of M. Gromov on it; see https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/carnot_caratheodory.pdf
It also arises in some applied area, namely, geometry of vision, on this I can not find exact reference now.