I don't know if this deserves to be an answer or a comment, as it includes information you surely know.

The terminology of matroid theory borrows heavily from graph theory, linear algebra, and other fields. A dependent set in a graphic matroid corresponds to a cycle in the underlying graph, so a general dependent set in a matroid is called a circuit. The length of the smallest cycle of a graph is its girth, so the same word is used for a general matroid. As you're well aware, the *spark* of a matrix is the *girth* of its corresponding vector matroid.

You're correct that your notion appear to be dual to spark. The appropriate terminology here could have been "cutset", as that's the corresponding notion in a (connected) graph, but for consistency it's *cogirth*. The *co-spark* of a matrix is the *cogirth* of its corresponding vector matroid.

A matroid is representable (ie, as a vector matroid) over a field if and only if its dual matroid is representable. Moreover, the transformation taking a representation of a vector matroid to a representation of its dual matroid is completely effective. It follows, in a sense, that studying the co-spark of a matrix is equivalent to studying the spark of another matrix. In particular, any computational hardness results for spark carry over to co-spark. It seems, then, that the notion of co-spark would be useful primary as terminology and as a way of getting a handle on things, but not actually for new ideas.

Here's one paper that has explicitly looked into these sorts of things:
On the (co)girth of a connected matroid.