According to Wikipedia, two main kinds of “cause” are used in (U.S.) legal assessments of liability: cause-in-fact and proximate cause. A cause-in-fact is anything without which something would not have happened. The same event could clearly have multiple causes-in-fact. Cause-in-fact is a necessary but not sufficient condition for proving proximate cause. Proximate causation involves the additional element that the causing event be sufficiently related to the injury for the courts to consider it “the” cause of the injury for purposes of liability. I think it serves effectively as a kind of model for talk about “the” cause of something in general.
The notion of proximate cause seems close to the naive notion of cause that Russell wanted to remind us plays no role in modern science, and at the same time to the intuition of causal efficacy that Whitehead took to be involved in the common-sense apprehension of medium-sized wholes. I have associated both of these with what I have called “causality in the modern sense”. Proximate causes differ from these insofar as the law is only concerned with the proximate causes of particular events or states of affairs, whereas Russell and Whitehead were both concerned with what are taken to be repeatable cases of causal efficacy.
It is important to point out that the notion of proximate cause is explicitly tied to questions of legal liability. In the wake of Kant and Brandom, it should not be surprising to find that more generally, the descriptive “causality in the modern sense” that allows us to reductively talk about “the” cause of something has this close connection to considerations of blame and culpability. Similarly, Aristotle’s “categories” are etymologically kinds of accusations, and Locke spoke of the person as a “forensic” concept.