Ring came from Zahlring, which was Hilbert's term for what we would essentially call a ring of algebraic integers. Dedekind earlier used the term ordnung (= order, taken from the Linnean classification terminology like class and genus). For more on this, see the comments to the question Why is "h" the notation for class numbers?.
Fields in the algebraic sense used to be called bodies (thus closer to French and German). [Edit: In 1900, Pierpoint's "Galois' Theory of Algebraic Equations, Part II", in the second volume of Annals of Mathematics, uses "body" for field and "inferior body" for subfield, introduced on page 25. In 1910, Legh Reid's "The elements of the theory of algebraic numbers" uses the term "realm" for field, or more specifically for number field. Reid's text can be found on Google books, and on p. vi of the preface he writes that "realm" is synonymous with Körper, corpus, campus, body, domain, and field. In 1934, Heilbronn and Linfoot wrote a paper "On the imaginary quadratic corpora of class-number one", so corpus was still in use in the early 1930s.]
You can find the original Lascoux & Schützenberger paper here. My French (especially mathematical French) is not great, so I haven't been able to determine how the term "plaxique" comes in. However, I can observe that L&S first introduce la congruence plaxique and define le monoïde plaxique as the quotient of the free monoid over the congruence. So, it seems to me that they were really thinking of the congruence as plactic/plaxic more than the monoid itself (perhaps a fine distinction?). They highlight the relevant properties of the congruence in Proposition 2.5, so maybe that provides a clue?
EDITED TO ADD: A quick scan of the OED yields no results for either "plactic" or "plaxic", but there is one result for the Latin "plaxus" under the etymology for the obsolete word "plash" (To bend down and interweave (stems partly cut through, branches, and twigs) so as to form a hedge or fence.):
an unattested post-classical Latin form *plaxus , alteration of classical Latin plexus , past participle of plectere to plait, interweave, twine (see plexus n.)
So, perhaps "plaxique" is meant to invoke a sense of intertwining or weaving? I could see how that could apply to the congruence relation.
Bonus fun fact: Plaxico Burress makes an appearance in the OED in a citation for the entry "return date":
New York Giants star receiver and gun nut Plaxico Burress breezed in and out of Manhattan Criminal Court in 15 minutes yesterday, with little happening besides the setting of a June 15 return date.
Best Answer
I think the OP referes to the modern meaning of the word, in which case, according to that website, it first appeared in german physicist Woldemar Voigt's paper Die fundamentalen physikalischen Eigenschaften der Krystalle in elementarer Darstellung published in 1898. (I do not have access to this paper, but probably this deals with deformation tensors in crystals).