Let $M$ be a differentiable manifold.

Is there a name for the maximum number of globally defined independent vector fields on $TM$ which are tangent to the fibers of $TM\to M$? Is there a name for the maximum number of globally defined independent vector fields on $TM$ which are tangent to the fibers of $TM\to M$ and whose mutual flows commute, i.e. they are vertical and have pairwise zero Lie bracket? What kind of characteristic classes can be used to compute such quantities? What are these maximum numbers for $M = S^n$?

Edit:(After the answer by Michael Albanese)

The vertical rank of $TM$ is the maximum number of independent commuting vertical vector fields on $TM$. the rank of $M$ is the maximum number of independent commuting vector fields on $M$?This terminology coined by Milnor

Question: Is the vertical rank of $TM$ equal to the rank of $M$?