Dear Nigel, causality is not an observable (quantity) with a value and a unit; so it is not identified with any operators.
Causality is a principle. In a broader scientific and colloquial context, causality is any property of the relationship between the cause and its effect.
However, in physics, we mean something more particular by causality. In classical physics, we mean the following proposition:
If a cause takes place at time $t_1$ and its effect at time $t_2$, then $t_1<t_2$ must hold.
In other words, the cause precedes its effect.
It's obvious that if the principle above would be violated, the world would become logically inconsistent. Events at time $t_1$ could cause some events at time $t_2$ which would generally cause different events at time $t_1$, producing contradictory answers to the question what happened at time $t_1$.
Looking at causality from a relativistic viewpoint
In the special theory of relativity, the statement above must still hold for the history of spacetime to be free of logical contradictions. However, special relativity is based on the principle of relativity that says
Laws of physics take the same form in all inertial frames - those that are in uniform motion relatively to one chosen inertial system.
This must be true for all laws, including the principle of causality itself.
If this principle of relativity is combined with the principle of locality above, we may actually derive a stronger statement. In relativity, the delay between two events depends on the inertial system: simultaneity of events is relative, we say. So two events may be chronologically ordered in the opposite way if you switch into a different inertial system. However, spacelike separated events remain spacelike separated events; and an event in the future (or past) light cone of another event stays in the same cone from the viewpoint of all inertial frames.
Applying the principle of relativity to the principle of causality, we may derive a stronger, relativistic principle of causality:
If a cause takes place at point $P_1$ in spacetime and if its effect takes place at point $P_2$ in spacetime, then $P_2$ must belong to the future light cone of $P_1$.
This is a stronger statement than the original one (about the ordering of $t_1$ and $t_2$): the relativistic causality implies the ordinary causality, but something more (it implies the non-relativistic condition from the viewpoint of all relativistic inertial frames). A cause is not only unable to affect its past, like in the non-relativistic causality; it is unable to affect the spacelike-separated points in the spacetime, too.
Any violation of the relativistic causality - which means that causes may only influence their future light cones - would lead to the same logical contradictions that I explained in the non-relativistic context. In particular, you wrote:
This merely shows that SR is formally inconsistent if the $v \leq c$ boundary condition is violated, doesn’t it; despite there being a narrative saying FTL travel violates causality?
Well, indeed. However, special relativity is demonstrably a valid theory of our spacetime (at least locally). So the "mere" inconsistency of special relativity that you mentioned, in a somewhat incomprehensibly dismissive tone, would automatically mean an inconsistency of the whole Universe which is a pretty serious problem. There's no doubt that there can't be any signals that move faster than light. Logical consistency is an omnipresent and unquestionable assumption in all of physics (and maths), so one is always allowed - and encouraged - to assume it. When we assume it, we may easily show that faster-than-light motion violates causality. In fact, relativistic causality is exactly what bans faster-than-light motion.
I am convinced that this text explains - and fully unmasks - all deeper and more foundational facts and arguments behind the notion of causality in physics.