The XY_link system is being used for three purposes.
First, NY_link is mandatory, since the default name "New York Supreme Court" is a name for something other than the highest court in New York, and therefore cannot be a redirect.
Second, XY_link for several other states/territories is used to link to the official name (or the version used in Wikipedia, for example, a superfluous "Commonwealth" is often omitted). However, WP is sometimes wrong (RI, AL, maybe others).
This is optional, though, since redirects are usable here. There is nothing wrong with relying on redirects, however, it's nice to have the correct name show up on mouse-overs.
Third, WA_link is used for "Washington Supreme Court", since the default is "Washington (state) Supreme Court". This is presumably because the parent template leaves the disambiguation linking in. (Things like "History of Washington (state)" have to be distinguished from "History of Washington, D.C.", say.) Georgia is the only other state with a similar issue, but in this case, the default suffix works.
Again, using this is optional, since the redirect works, and again, it's nice to have have the correct name show up on mouse-overs. Choor monster (talk) 15:33, 28 March 2014 (UTC)