I'm trying to learn tensor calculus on my own, and I thought it would help if I could get some historical perspective, like who developed the component-free approach to tensors, and around what time it was developed.
Roger Penrose has something to say about tensor calculus in his popular book "The Road to Reality". He discerns a "mathematician's notation" and a "physicists's notation" (which is "abstract index calculus" and his invention) plus gives a graphical notation (also his invention).
The best quick introduction to abstract index calculus is R.Wald's General Relativity. Roughly speaking it is just a different reading of component-notation tensor calculus, or, Ricci calculus.
The historical landmark book on Ricci calculus is Tullio Levi-Civita's The Absolute Differential Calculus. First english edition 1926. Levi-Civita was a student of Gregorio Ricci-Curbastro, who developed this calculus 1887-1896. A nice quick introduction into what E.Cartan called "the debauch of indices" is E.Schrödinger's Space-Time Structure.
The component-free (but not mess-free!) "mathematician's notation" $\nabla_X$ is due to J.L.Koszul, first presented by himself in his famous "Lectures on Fibre Bundles and Differential Geometry" at the Tata Institute Bombay 1960. (I have the 1965 issue, one of my most beloved math books.) But Koszul's calculus was first publicly used in a paper by Nomizu: "Invariant affine connections on homogenous spaces", Amer.J.Math. 76 (1954). -- This is the history according to M.Berger: "A Panoramic View of Riemannian Geometry".
More history is in Kobayashi&Nomizu Vol.I p.287 - but they don't mention Koszul!
Last but not least, here is the best mathematical book on the history of the subject:
M.Spivak, A Comprehensive Introduction to Differential Geometry Vol.II
with translations of historical texts and maths from Gauss to Riemann and a comparative presentation of all diverse tensor calculi from Ricci, Cartan, Koszul, ending with Ehresmann connections in principal fibre bundles. Except abstract index calculus. He also attributes the $\nabla_X$ notation to Koszul, but doesn't quote his landmark Tata lectures.
I find abstract index calculus superior to Koszul's for a few reasons, e.g. a) Koszul's is a partial directional derivative b) stuff tends to get presented upside-down (compare the definitions of symmetric connection) and c) left-right messed up (higher covariant derivatives) and d) due to a) there are lots of superfluous Lie brackets (e.g. definition of curvature and torsion)
Crazily, I'm currently writing up my own tensor calculus, which is abstract index calculus with them abstract indices replaced by abstract multilinear algebra, and bridling the horse from the right end.