The *notions* of Shelah cardinals and Woodin cardinals 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. To get an idea of how quickly the term was adopted, already the Shelah-Woodin paper mentions it, see page 384 and Definition 4.1 in page 392: 
> We define here two large cardinals: $\mathrm{Pr}_a(\lambda,f)$, $\mathrm{Pr}_a(\lambda)$ by Shelah (Definition 3.5) and $\mathrm{Pr}_b(\lambda)$ by Woodin -- now called a Woodin cardinal. 

Note that, in spite of its publication date, the results of the paper were obtained quickly after the results in the Martin's Maximum paper, itself published in 1988. In the $\mathsf{MM}$ paper, we read (page 27)
> Woodin, aware of [Foreman's work on $\mathfrak c$-dense, normal, fine ideals on $[(2^{\aleph_0})^+]^{\aleph_1}$] and of Theorem 12 [that $\mathsf{MM}$ implies the saturation of the nonstationary ideal], proved the following proposition. It was proved simultaneously with the third author's realization that this technique of $S$-complete forcing could be used together with the results of Sections 1 and 2 to prove Theorem 21 [on versions of $\mathsf{MM}$ comaptible with $\mathsf{CH}$]. In a phone call to the first author, Woodin, unaware of Theorem 21 and its consequences, announced his proposition.

Woodin's result indicates how to prove from large cardinals that $L(\mathbb R)$ is (elementarily equivalent) to the $L(\mathbb R)$ of a Solovay's model. Working on optimizing this result led to the notions of Shelah and Woodin cardinals. 

The relevance of Woodiness was quickly seen in the context of homogeneously Suslin sets, relevant to determinacy, which led to the immediate adoption of the notion.

The first appearance of the *term* in the literature 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 cardinals" and "Shelah cardinals" are probably due to them, but due to the influence of the concept, the terms were in use, particularly Woodin cardinals, before the papers appeared. 



  [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