There is no such thing as an unambiguous "speed of causality", because causality itself is a very vague notion when you actually try to nail it down.
What is true is that the speed that we call the speed of light - but is, as you say, actually meant to be the speed of all massless particles - limits information transfer in the following sense:
For any event $x$ (i.e a time and a place) in spacetime, there is a set of events (the past lightcone and the events inside it) which can "causally" influence what happens at the event. The light cone is precisely the set of events from which something travelling at the speed of light can reach $x$ - from any event outside of it you would need to travel faster-than-light.
Since nothing - massless or massive - can travel faster than the speed of light, you might be tempted to say that therefore, of course, the speed of light is "the speed of causality"; how could anything influence something else without travelling? But the problem is, again, that our intuitive notion of what it means to be "cause and effect" - or to have "causality" - doesn't really map neatly to the physical ideas of something travelling, nor does the light-cone picture of causality alone really produce a world of "cause and effect" that we would like.
On the one hand, without travelling faster-than-light, it is possible to imagine spacetimes with so-called closed timelike curves, and something travelling along such a curve is in its own past lightcone. Cause without effect, or rather an effect that is its own cause. Is this "causality"? What is the "speed of causality" in this case? (The common answer is that such spacetimes are bad because they "violate causality".)
On the other hand, quantum theory makes everything even more complicated (as usual) - Bell's theorem tells us that either there are "effects" that propagate superluminally, or the world is not realist (for more discussion of this, see this answer of mine and this answer of mine). Crucially, which of these two to choose is a realm of metaphysics called quantum interpretations, but the predictions of quantum mechanics do not (or only in extremely contrived cases, depending on who you listen to) depend on the interpretation chosen. And so, the "speed of causality" - indeed, perhaps causality itself - is exposed as the incoherent idea that it is: In some interpretations of quantum mechanics, this speed is infinite - measurements on "one part" of a wavefunction instantaneously affect every part of this wavefunction, throughout the whole universe - and in others it is still finite, and a measurement doesn't actually have to propagate any changes at all, and there's probably all sorts of hybrid interpretations, and yet it doesn't make one bit of difference to their predictions of the world we observe.
For further ruminations on the incoherence of the idea of causation, I recommend Norton's (in)famous paper "Causation as Folk Science".