I think the definition of transversality of two smooth maps with common codomain is clear to me - we want the intersection of the images to be nondegenerate in that not only are the images not tangent, but moreover their tangent spaces actually cover (sum to) the tangent space of the ambient manifold. In this binary case, it's easy to show tranversality of two vector spaces is equivalent to the codimension of the intersection equalling the sum of their codimensions.
I am confused as to what transversality of a family of submanifolds should even mean. Some authors define a finite family of vector spaces to be mutually transverse if the codimension of the intersection is the sum of the codimensions, and this is a straightforward generalization, but I would like to understand why it's also the correct one.
For instance, why don't we want to say that three curves in $\mathbb R^3$ which intersect in a pairwise non-tangent manner are transverse at their intersection? The codimension definition makes this impossible since the codimension of each curve is two, so the sum is six while the maximal codimension of a subspace is three.
Added. Just to make sure, since I suddenly can't find a definition anywhere: do we say a family of vector subspaces $V_1,\dots ,V_n\leq V$ is mutually transversal if $$\operatorname{codim} \bigcap_{i\in [n]}V_i=\sum_{i\in [n]}\operatorname{codim}V_i,$$ or $$\operatorname{codim} \bigcap_{i\in J}V_i=\sum_{i\in J}\operatorname{codim}V_i\text{ for all }J\subset [n]?$$