The first appearance of the term is in the papers by Donald A. Martin and John R. Steel, 
> [*Projective determinacy*][1]. Proc. Nat. Acad. Sci. U.S.A., **85 (18)**, (1988), 6582–6586. [MR0959109 (89m:03041)][2],

and  
> [*A proof of projective determinacy*][3]. Journal of the American Mathematical Society, **2 (1)**, 1989, 71-125. [MR0955605 (89m:03042)][4].

Both terms Woodin and Shelah for large cardinals are probably due to them, but due to the immediate influence of the concept, the terms were in use, particularly Woodin cardinals, before the paper appeared. The *notions* themselves were introduced by Shelah and Woodin in their joint paper
> *Large cardinals imply that every reasonably definable set of reals is Lebesgue measurable*. Israel J. Math., **70 (3)**, (1990), 381–394. [MR1074499 (92m:03087)][5],

which itself was the result of the hugely influential Martin's Maximum paper by Foreman, Magidor, and Shelah. In their "Lebesgue measurable" paper, Shelah cardinals are those $\lambda$ that satisfy property $\mathrm{Pr}_a(\lambda)$, and Woodin cardinals are those that satisfy $\mathrm{Pr}_b(\lambda)$. Overall, the current notation seems better.


  [1]: http://www.pnas.org/content/85/18/6582
  [2]: http://www.ams.org/mathscinet-getitem?mr=959109
  [3]: http://www.ams.org/journals/jams/1989-02-01/S0894-0347-1989-0955605-X/S0894-0347-1989-0955605-X.pdf
  [4]: http://www.ams.org/mathscinet-getitem?mr=955605
  [5]: http://www.ams.org/mathscinet-getitem?mr=1074499