By a relativistic theory, I mean one where motion is relative, and there is no action-at-a-distance. A more precise definition is a theory with Lorentz covariance, but that concept was not understood in Maxwell's lifetime. Motion is relative in Newtonian, Lagrangian, and Hamiltonian mechanics, but action-at-a-distance was necessary to explain gravity.
Here is what Maxwell said in an elementary textbook, Matter and Motion (1876):
102. RELATIVITY OF DYNAMICAL KNOWLEDGESince then, the cosmic microwave background (CMB) radiation has been discovered to be a landmark for motion in space. Our galaxy is moving at 627±22 km/s relative to the CMB rest frame.
Our whole progress up to this point may be described as a gradual development of the doctrine of relativity of all physical phenomena. Position we must evidently acknowledge to be relative, for we cannot describe the position of a body in any terms which do not express relation. The ordinary language about motion and rest does not so completely exclude the notion of their being measured absolutely, but the reason of this is, that in our ordinary language we tacitly assume that the earth is at rest.
As our ideas of space and motion become clearer, we come to see how the whole body of dynamical doctrine hangs together in one consistent system.
Our primitive notion may have been that to know absolutely where we are, and in what direction we are going, are essential elements of our knowledge as conscious beings.
But this notion, though undoubtedly held by many wise men in ancient times, has been gradually dispelled from the minds of students of physics.
There are no landmarks in space; one portion of space is exactly like every other portion, so that we cannot tell where we are. We are, as it were, on an unruffled sea, without stars, compass, soundings, wind, or tide, and we cannot tell in what direction we are going. We have no log which we can cast out to take a dead reckoning by; we may compute our rate of motion with respect to the neighbouring bodies, but we do not know how these bodies may be moving in space.
We cannot even tell what force may be acting on us; we can only tell the difference between the force acting on one thing and that acting on another.
Relativity books sometimes make fun of Maxwell because he wrote the 1878 9th Ed. Encyclopædia Britannica article on the aether. He died in 1879, long before the discovery of the Lorentz transformation. But I think the essay holds up well as a summary of scientific knowledge of the aether at the time. That essay concluded:
No theory of the constitution of the aether has yet been invented which will account for such a system ... Whatever difficulties we may have in forming a consistent idea of the constitution of the aether, there can be no doubt that the interplanetary and interstellar spaces are not empty, but are occupied by a material substance or body, which is certainly the largest, and probably the most uniform body of which we have any knowledge.The uniformity of the aether is the essence of relativity. That uniformity makes light and electromagnetism the same for all observers.
Einstein did not like the term "relativity" initially, but Poincare and others used the term before Einstein wrote about the subject, and the term caught on from them.