What do physicists mean by "information"?
In the case of relativity, "information" refers to a signal that enforces causality. That is, if event A causes event B, then some signal must travel from A to B. Otherwise, how would B "know" that A had occurred?
Some examples:
- Light (signal) from a candle (A) hits your eye (B), causing you to see it.
- Electricity (signal) flows from a connected switch (A) to a light bulb (B), turning it on.
- Your friend (A) throws a wad of paper (signal) that hits you (B) in the back of the head, causing you to turn around to see who's trying to get your attention.
In all of these cases, the effect (B) comes after the cause (A) because there must be some signal from A that interacts with B to cause B to happen. The technical term for this is "locality." Over the centuries of studying how the universe works, scientists have found that all causes are local to their effects; nothing happens at a distance without something (light, sound, matter, etc.) acting as a go-between. [1] If you want to interact with some distant object (a friend, a planet, an enemy target), you either have to go there yourself or send something in your place (a letter, a satellite, a missile).
Let's consider the case of a laser beam swept across the face of the Moon. Let's further imagine that there are two astronauts, Alice and Bob, on the surface of the Moon with a large distance between them. The laser spot sweeps across the Moon and falls upon both Alice and later Bob, with the spot moving at faster than the speed of light. So, the question is, does that spot constitute a causality signal from Alice to Bob? The answer is no, because nothing Alice does will affect how the spot moves or when it moves or even if it moves. The cause of the light is on Earth and is not local to Alice. Nothing Alice does will change the spot that Bob sees.
There is a way that Alice can use the spot. She can hold up a mirror and reflect the laser beam towards Bob. The reflected laser beam is a causality signal because its origin is local to Alice. Alice can choose whether or not to reflect the beam at Bob. But, notice that this signal travels at the speed of light. It will arrive after the laser spot sweeping across the surface.
[1] This is why Einstein and others objected to quantum entanglement weirdness. It looks like signaling at a distance, but it's really not. Various mathematical and experimental discoveries show that not even entanglement's "spooky action at a distance" can transmit information faster than light. Quantum teleportation has been demonstrated in the lab, but there must be a slower-than-light signal between the sender and receiver to make the system work. There's far too much detail to go into here.
In the context of relativity and nonproagation of information at greater than lightspeed between two separated points $A$ and $B$, "information" simply means any particle, feature of in a field (EM, quantum field, curvature in spacetime ...), message or so forth that could allow a causal link between $A$ and $B$, i.e. could make $B$'s physics depend on $A$'s presence (and contrariwise).
Therefore we know, for example, that, in relativistic limits, the wonted heat diffusion equation $(\partial_t-k\,\nabla^2)T=0$ cannot be correct, for its solution in 1D is a superposition of heat kernels $\frac{1}{\sqrt{4\,\pi\,k\,t}}\exp\left(-\frac{x^2}{4\,\pi\,k\,t}\right)$. Suppose $A$ sits at $x=0$ and imparts an impulse of heat at $x=0$ (i.e. heats a tiny region near $x=0$ intensely and quickly) and $B$ at $x=L$ has agreed to raise a flag as soon as $B$ senses a rise in the temperature at $x=L$. The time dependence of the heat kernel, we see that the temperature begins to rise at $L$ at $t=0$, so the signalling speed in this case between $A$ and $B$ would be arbitrarily fast and limited only by the signal to noise ratio of $B$'s measurement.
Sometimes it is stated that the signal, or "information propagation" speed in a dispersive medium is the group velocity, since this is approximately the speed at which any narrowband modulation is propagated on a carrier light wave and indeed this does seem to impose speed limits of $<c$ when applied to regions of anomalous dispersion in optical mediums. But this is only an approximation which breaks down for very wideband signals. Ultimately one needs to return to basic causality limits, e.g. as found by the Paley-Wiener criterion, to work out what limitations there must be on things like optical dispersion.