Shouldn't "speed of light" really be "speed of electromagnetic waves"? Since all electromagnetic waves travel at the speed of light ... shouldn't its name be Speed of Electromagnetic wave? 
 A: The speed of light is a geometrical property of the flat spacetime. See What is so special about speed of light in vacuum?  and also Special Relativity Second Postulate for more on this. Any massless object necessarily travels at this speed, so it applies not only to electromagnetic radiation but also to gravitational waves.
Describing $c$ as the speed of light is largely historical since light was only understood to be electromagnetic radiation after Maxwell's work, and of course the theory of gravitational waves had to wait for Einstein. Nevertheless the description is perfectly accurate since the speed of light is indeed $c$, and the description is both more succinct and elegant that the propagation speed of any massless object.
A: As an experimentalist, It depends on your detector! 
Everything that shines your detector is light
Sometimes that detector is an eagle eye or a special fish eye that can see UV light.
Sometimes is a human eye for visible light
And sometimes a night camera for IR light or a radar to see the depth of the cosmic with radio frequency light. Or as others mentioned a gravitational wave that shines the LIGO too.
A: The term "speed of light" means the speed of electromagnetic radiation in space free of matter and other none-zero point fields also called the quantum electrodynamic (QED) vacuum. It is also the maximum speed of energy-mass transfer in an inertial reference frame. Since 1983 it has been defined to have the exact value c = 299792458 m/s. As John Rennie mentioned, the term "speed of light" is a historical carryover that takes on a richer meaning in modern theories. 
A: Of course John Rennie is correct, but I would like to add a few things.


*

*the classical EM wave is made up of a herd of photons, and the QM description works great together with the classical

*photons always travel at speed c in vacuum, when measured locally

*the speed of EM waves is c in vacuum, when measured locally

*EM waves have a speed in denser media less then c, because the wavefront slows down, but individual photons still travel at speed c, because they always travel in the vacuum between atoms in the material

*this is why historically it is called speed of light

*gravitational waves travel at the same speed c, in vacuum , when measured locally

*anything with no rest mass as per SR will always travel at this speed c in vacuum, when measure locally

*there are other particles, like the gluon, that have no rest mass, and should travel at this speed, but these are in confinement
The theoretical gluons do travel at the same speed c. This speed c is the only speed in the universe, but light (and GWs) do not move in the time dimension, light does not experience time the way we do. To experience time, you need to gain rest mass, and if you do gain rest mass, you will start moving in the time dimension, and start experiencing time.
To think about this in an easy way is to say that the fabric of spacetime is so, that the excitation in the fields (EM and GW) can travel at this speed. Today we know that these waves are made up of massless particles (photon for EM and gluon for GW). It is the fabric of spacetime that determines the speed of these waves.
