In physics, as Emmy Noether showed us with her beautiful theorem, invariance turns out to entail the conservation of energy and other bedrock conservation principles — "a fact," noted Richard Feynman, "that most physicists still find somewhat staggering."No, Einstein did not conceive the geometrization of gravity. As a recent paper noted:
And in the mind of Albert Einstein, the idea of invariance led first to e = mc2, and then to the geometrization of gravity.
So why aren't we hearing constantly about Einstein's theory of invariance? Well, "invariant theory" is what he later said he wished he had called it. And that's what it should have been called, since invariance is its very essence. The speed of light, the laws of physics are the same for all observers. They're objective, absolute — invariant. Simultaneity is relative, unreal.
But no. Einstein had to go and talk about the "principle of relativity." So "relativity"—and not its opposite, "invariance"—is what his revolutionary theory ended up getting labeled.
It is generally believed that Einstein identified gravitation with the non-Euclidean geometry of spacetime. However, contrary to common belief, as Lehmkuhl showed , Einstein himself did not believe that general relativity geometrized gravitation: "I do not agree with the idea that the general theory of relativity is geometrizing Physics or the gravitational field" .Einstein also did not originate the terms "relativity" or "principle of relativity". As well as I can determine, Maxwell invented the term "relativity", and Poincare popularized the "principle of relativity", long before Einstein ever wrote anything on the subject.
 D. Lehmkuhl, Why Einstein did not believe that General Relativity geometrizes gravity. Studies in History and Philosophy of Physics, Volume 46, May 2014, pp. 316-326.
 A letter from Einstein to Lincoln Barnett from June 19, 1948; quoted in .
In short, relativity was named by those who invented it, not Einstein.
Holt likes "invariance" because it suggests a group action, but Einstein missed that. The group is called the "Lorentz group", because that is what Poincare called it in a 1905 paper that was published before Einstein submitted his famous relativity paper. Poincare named it after Lorentz, who did pioneering work on it ten years earlier.
I don't see why "invariance" is a better name anyway. The theory's origin was with attempts to understand the motion of the Earth, with a key observation being that the motion we see is relative to the Earth's frame of reference. The theory shed new light on an ancient and easy-to-understand question. Getting to invariants is a little more obscure.